Calculating circuit for error correction

ABSTRACT

A Euclidean mutual division circuit that has (2t+1) calculation units and (2t+3) registers for error correction where t is the number of symbols that can be error-corrected. The division unit and each calculation unit conduct Euclidean mutual divisions of either a normal-connection, a cross-connection, or a shift-connection division as directed by a control unit. The final calculation unit outputs a value to a division unit which divides it by another value, then the result of the division is returned to each of the calculation units. Registers that store the inputs for the Euclidean mutual division method include A-side and B-side registers. A-side registers store the coefficients of polynomials Qi(X) and λi(X), and B-side registers store those of Ri(X) and μi(X). The Euclidean mutual division circuit has a reduced circuit scale for high-speed operation and for increased throughput.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a calculating circuit for error correction, and more particularly to a multiple division circuit for carrying out Euclidean mutual division.

2. Description of the Prior Art

In the implementation of an error-correcting system using an error-correcting code represented by a Bose-Chaundhuri-Hocquenghem (BCH) code or a Reed-Solomon code, a means for determining an error locator polynomial from a syndrome generated from a received signal plays the most important role (see "7R-C601-018(4740) Hardware implementation of a high-speed multiple error-correcting circuit using the Reed-Solomon code (1)").

One well known process for determining such an error locator polynomial uses an Euclidean mutual division algorithm (see "7R-C601-020(4788) Hardware implementation of a high-speed multiple error-correcting circuit using the Reed-Solomon code (2)").

The Euclidean mutual division method is generally known as an algorithm for obtaining the most common divisors of two polynomials. In the error-correcting code, an error locator polynomial can be calculated from a syndrome by the skillful application of a calculation using the Euclidean mutual division method.

The calculation using the Euclidean mutual division method is advantageous in that it can be composed of a systolic-array architecture which can be realized by cascading a plurality of relatively simple calculation units (hereinafter referred to as "mutual division units").

Howard M. Shao, et. al. have proposed an example in which the Euclidean mutual division algorithm is implemented by a systolic-array architecture (see Howard M. Shao, et. al. "A VSLI Design of a Pipeline Reed-Solomon Decoder" IEEE Trans. on Computers Vol. C-34, May 1985). This method will hereinafter be referred to as a "method A". The method A however has defects in that its algorithm is not perfect and that each unit needs two multipliers for a finite field. Thus, when a system that needs high-speed real-time processing is to be constructed based on this method A, its circuit scale is increased.

We have previously proposed a method as an improvement of the method A as disclosed in U.S. patent application Ser. No. 07/623,235 filed Dec. 6, 1990. This proposed method will hereinafter be referred to as a "method B". The method B employs a revised Euclidean mutual division method (2) as disclosed in the above article by Howard M. Shao et. al., and replaces two finite field multipliers in the mutual division unit with a multiplier for a finite field and a divider for a finite field. Furthermore, in the method B, the finite field dividers used in a plurality of cascaded mutual division units are shared by a single divider for a finite field. The circuit arrangement used to implement the method B is therefore reduced in circuit scale.

We also have proposed an improved method (hereinafter referred to as a "method C") which is entirely different from the method B and employs only one additional circuit for sharing the divider and only one degree control circuit, as disclosed in Japanese patent application No. 3-254,183 filed Sep. 6, 1991. According to the method C, a circuit for sharing the divider and a control circuit therefor are not required because data to be divided can be extracted from one location at all times. In addition, a circuit for judging operation from the degree of a polynomial and a control circuit are realized by single circuits, respectively. Therefore, the method C can be implemented by a circuit arrangement of reduced circuit scale.

Now, the execution of only one calculation by one calculation unit in one step according to the revised Euclidean mutual division method (2) will be considered below. That is, the number of calculation units that are essentially required except when two or more calculations are carried out by one calculation unit through multiplexing such as time-division multiplexing will be considered below.

The numbers of calculation units required by the respective methods A, B, C are 8t, 4t, 4t-1, respectively. The numbers of calculation units required by the methods B and C are reduced to half the number of calculation units required by the method A. This is because one divider, rather than two multipliers, is used by each of the units, thereby reducing the number of multipliers required by respective calculation units to one.

While the number of calculations (multiplications) needed by the algorithm itself of the revised Euclidean mutual division method (2) is only 2t for one step, the circuit arrangement which implements the methods B and C requires as many calculation units as twice the number that is essentially required by the algorithm, and hence is wasteful. Similarly, the number of registers required to store the coefficients of polynomials is about twice the number that is essentially required by the algorithm.

Error-Correcting Procedure

An error-correcting procedure will be described below.

An error-correcting system having a code length n which can error-correct t symbols using a finite field GF(2^(m)) will be considered. It is assumed that a jth error location as counted from the starting end of the code, which is selected to be the 0th position, is expressed by α^(j). If the code contains a total of m errors, then the entire errors of the code can be described by an error location X_(i) (i=l, . . ., m) and an error pattern Y_(i) (i=l, . . ., m). Therefore, if there are m (m≦2t) errors in total, then m sets of (X_(i), Y_(i)) are required.

A polynomial known as an error locator polynomial σ(X) is determined using the error location X_(i) (i=l, . . . m). The error locator polynomial (σ(X) becomes 0 if the error position X=X_(i) l (i=l, . . ., m) . The error locator polynomial σ(X) is expressed by the following equation (1): ##EQU1## The coefficients of the error locator polynomial σ(X) are expressed as follows:

    σ(X)=1+σ.sub.1 X+σ.sub.1 X+σ.sub.2 X.sup.2. . .+σ.sub.m X.sup.m                                   ( 2)

Using the error locator polynomial σ(X) and a syndrome polynomial S(X), an error evaluator polynomial ω(X) is defined by:

    ω(X)=S(X)σ(X) (mod X.sup.2t)                   (3)

The error-correcting process will next be described below successively with respect to the following steps:

STEP 1

2t syndromes S, expressed by the following equation (4):

    S=[S.sub.1, S.sub.2, . . ., S.sub.2t ].sup.T               ( 4)

are determined as the product of a received signal r and a parity-check matrix H, and hence are given as follows:

    X=Hr                                                       (5)

Then, a polynomial called a syndrome polynomial S(X) having the syndromes S thus determined as the polynomial coefficients is defined as follows: ##EQU2##

STEP 2

An error locator polynomial σ(X) and an error evaluator polynomial ω(X) are determined from the syndrome polynomial S(X), using the algorithm of the Euclidean mutual division method.

STEP 3

The error location X=X_(i) ⁻¹ (i=l, . . ., m) is searched. Using the coefficients of the error locator polynomial σ(X) that have been determined in the step 2, all elements X=α⁰. . .α^(n-1) contained in the finite field GF(2^(m)) are substituted in the error locator polynomial σ(X), and the location where σ(X)=0 is determined as the error location X_(i) (i=l, . . ., m).

STEP 4

Once the error location X_(i) (i=l, . . ., m) is determined, the error pattern Y_(i) (i=l, . . ., m) is calculated using the error evaluator polynomial ω(X) as follows: ##EQU3##

STEP 5

The received signal is corrected using the error location X_(i) (i=l, . . ., m) and the error pattern Y_(i) (i=l, . . ., m) .

The error correction is carried out in the steps 1 through 5 described above. For more details, reference should be made to the article "7R-C601-018(4740) Hardware implementation of a high-speed multiple error-correcting circuit using the Reed-Solomon code (1)" the article "7R-C601-020(4788) Hardware implementation of a high-speed multiple error-correcting circuit using the Reed-Solomon code (2)", and an article "7R-C601-021 (4817) Hardware implementation of a high-speed multiple error-correcting circuit using the Reed-Solomon code (3)".

Deriving an Error Locator Polynomial Using the Euclidean Mutual Division Method

A process of deriving an error locator polynomial will be described below. This process corresponds to the step 2 in the above error-correcting process. A method of determining the error locator polynomial σ(X) from the syndrome polynomial S(X) using the algorithm of the Euclidean mutual division method is known as follows:

Method of Deriving an Error Locator Polynomial

It is assumed that r₋₁ (X)=X^(2t) and r₀ (X)=S(X). Since the degree of S(X) is 2t-1, deg (r_(o) (X))<deg (r₋₁ (X)). Using r₋₁ (X) and r₀ (X), a division to determine a quotient which is a polynomial q_(i) (X) is repeated. The division is the same calculation as the Euclidean mutual division method, and is stopped when the following condition is satisfied: ##EQU4## where deg(r_(j) (X))≦t-1. At this time, r_(j) (X) can be expressed according to the equation given below by substituting the initially defined equations r₋₁ (X)=X^(2t) and r₀ (X) =S(X) successively in the equations r₁ (X) , r₂ (X), . . ., r_(j) ₃₁ 1 (X) from below which have been obtained in the above division process:

    r.sub.j (X)=S(X)A(X)+X.sup.2t B(X)

r_(j) (X) and A(X) obtained at this time become ω(X) and σ(X), respectively.

In order to achieve the above process through hardware, it is of importance to consider how divisions are successively carried out to determine q_(j) (X), r_(j) (X) and also how a process of inversely carrying out successive substitutions is performed in order to determine σ(X) from the equations r₁ (X), r₂ (X), . . ., r_(j-1) (X) that have been obtained.

In the Euclidean mutual division method, the degree of a residue as a result of a division may sometimes drop by two or more. Any hardware arrangement for implementing the Euclidean mutual division method must operate without fail in such an instance.

The above procedure can be achieved using a systematic algorithm, given below, by rewriting the above normal Euclidean mutual division algorithm such that the degree is reduced one by one. The systematic algorithm is a modification made by the applicant of the algorithm proposed in the literature Howard M. Shao, et. al. "A VSLI Design of a Pipeline Reed-Solomon Decoder" IEEE Trans. on Computers Vol. C-34, May 1985, and corresponds to a slight improvement of the revised Euclidean mutual division method (2) described in the article "7R-C601-020(4788) Hardware implementation of a high-speed multiple error-correcting circuit using the Reed-Solomon code (2)".

Revised Euclidean Mutual Division Method (2) (Initial Condition)

    R.sub.o (X)=S(X), Q.sub.o (X)=X.sup.2t

    λ.sub.o (X)=1, μ.sub.o (X)=0

    dR.sub.o =2t-1, dQ.sub.0 =2t                               (10)

(Repetition)

In an ith step,

    1.sub.i-1 (X)=dR.sub.i- 1-dQ.sub.i-1

the coefficient of degree dR_(i-1) of R_(i-1) (X) is a_(i-1) and the (dQ_(i-1))th coefficient of Q_(i-1) (X) is b_(i-1). . . (11).

[1] In case l_(i-1) ≧0 (normal mode),

    R.sub.i (X)=R.sub.i-1 (X)+(a.sub.i-1 /b.sub.i-1)Q.sub.i-1 (X).X.sup.li-1

    λ.sub.i (X)=λ.sub.i-1 (X)+(a.sub.i-1 /b.sub.i-1)μ.sub.i-1 (X).X.sup.li-1

    Q.sub.i (X)=Q.sub.i-1 (X)

    μ.sub.i (X)=μ.sub.i-1 (X)

    dR.sub.i =dR.sub.i-1 -1

    dQ.sub.i =dQ.sub.i-1                                       ( 12)

[2] In case l_(i-1) <0, a_(i-1) ≠0 (cross mode),

    R.sub.i (X)=Q.sub.i-1 (X)+(a.sub.i-1 /b.sub.i-1)R.sub.i-1 (X).X.sup.-li-1

    λ.sub.i (X)=μ.sub.i-1 (X)+(a.sub.i-1 /b.sub.i-1)λ.sub.i-1 (X).X.sup.-li-1

    Q.sub.i (X)=R.sub.i-1 (X)

    μ.sub.i (X)=λ.sub.i-1 (X)

    dR.sub.i =dQ.sub.i-1 -1

    d.sub.Qi =dR.sub.i-1                                       ( 13)

[3] In case l_(i-1) <0, a_(i-1) =0 (shift mode),

    R.sub.i (X)=R.sub.i-1 (X)

    λ.sub.i (X)=λ.sub.i-1 (X)

    Q.sub.i (X)=Q.sub.i-1 (X)

    μ.sub.i (X)=μ.sub.i-1 (X)

    dR.sub.i =dR.sub.i-1 -1

    dQ.sub.i =dQ.sub.i-1                                       ( 14)

(Stop Condition)

The process is stopped when it is repeated 2t times.

    i=2t                                                       (15)

(Result)

    σ(X)=ρ.sub.2t (X)

    ω(X)=R.sub.2t (X)                                    (16)

If the calculation is necessarily repeated by 2t steps as the stop condition, dR_(2t) which is finally obtained is indicative of the degree of ω(X). Naturally, the degree of σ(X) is R_(2t) (X)+1. According to this algorithm, σ(X) and ω(X) can be determined after the calculations in the 2t steps.

As described in the article "7R-C601-020(4788) Hardware implementation of a high-speed multiple error-correcting circuit using the Reed-Solomon code (2)", dR_(2t) eventually represents the degree of R_(2t) (X) , i.e., ω(X). When calculations are carried out in the normal and cross modes, dR_(i) and dQ_(i) in the calculating process are indicative of the degrees of R_(i) (X) and Q_(i) (X) , respectively. However, they do not represent degree in the shift mode in which the high-order coefficient of R_(i) (X) is 0. This is caused by the fact that even when the degree of a residue of one division drops by two or more in the Euclidean mutual division method, dR_(i) decreases one by one according to the algorithm of the revised Euclidean mutual division method (2). For a detailed example, reference should be made to the article "7R-C601-020(4788) Hardware implementation of a high-speed multiple error-correcting circuit using the Reed-Solomon code (2)".

Hardware Implementation of the Conventional Euclidean Mutual Division Algorithm

An example in which the algorithm of the Euclidean mutual division method is hardware-implemented using a systolic-array architecture is disclosed in the literature Howard M. Shao, et. al. "A VSLI Design of a Pipeline Reed-Solomon Decoder" IEEE Trans. on Computers Vol. C-34, May 1985 (method A) .

However, the disclosed hardware does not cope with the reduction by two or more of the degree of a residue in the division according to the Euclidean mutual division method, and does not fully realize the Euclidean mutual division algorithm.

The method A requires two finite-field multipliers in each unit, and hence results in a large circuit scale when a system capable of high-speed real-time processing is to be constructed.

The method B referred to above has been proposed by the inventor as an improvement of the method A. According to the method B, two finite-field multipliers in a mutual division unit are replace with one multiplier for a finite field and one divider for a finite field, and finite-field dividers in a plurality of cascaded mutual division units are shared by one divider for a finite field. In this manner, a circuit arrangement for carrying out the method B is reduced in circuit scale.

The method B employs a calculation unit (referred to as a "mutual division unit") 101 as shown in FIG. 1 of the accompanying drawings. The mutual division unit 101 carries out one step of the revised Euclidean mutual division method (2) described above. The four polynomials R_(i-1) (X) , Q_(i-1) (X) , λ_(i-1) (X) , and μ_(i-1) (X) as inputs in the steps of the Euclidean mutual division method are input successively from the coefficient of degree dQ_(i) -1. SF represents a flag indicative of the first coefficient. At the same time, dR_(i-1) and dQ_(i-1) are input to the mutual division unit 101.

The mutual division unit 101 includes data path switchers indicated by the dotted lines. When calculations are carried out in the cross mode, the data path switchers select crossed data. In the other modes, the data path switches select data which are not crossed. Specifically, when the coefficient of degree dR_(i-1) is input, i.e., when the SF flag is input, it is determined whether the coefficient a_(i-1) of degree dR_(i-1) of R_(i-1) (X) is 0 or not, and only if a_(i-1) ≠0, dR_(i-1) <dQ_(i) -1, the data are crossed by the data path switchers.

As a consequence, in the cross mode, a_(i-1) /b_(i-1) is held in a register 102, and in the other modes, b_(i-1) /a_(i-1) is held in the register 102, with calculations effected on all the coefficients of polynomials. Each time input data passes once through the mutual division unit, the mutual division unit executes one step of the revised Euclidean mutual division method (2). In the case where 2t mutual division units are cascaded, and inputs expressed by:

    R.sub.o (X)=S(X), Q.sub.o (X)=X.sup.2t

    λ.sub.o (X)=1, μ.sub.o (X)=0

    dR.sub.o =2t-1, dQ.sub.0 =2t                               (17)

are supplied, as shown in FIG. 2 of the accompanying drawings, the (2t)th mutual division unit 101 produces outputs σ(X), ω(X).

Using an example shown in the article "7R-C601-020(4788) Hardware implementation of a high-speed multiple error-correcting circuit using the Reed-Solomon code (2)", operation will be described by way of example with reference to FIGS. 3 through 6 of the accompanying drawings. Calculations are carried out in, the cross mode in FIGS. 3 and 5, and in the normal mode in FIGS. 4 and 6. As shown in FIG. 6, it can be seen that σ(X), ω(X) are determined as a result of the passage of data through the mutual division unit 101 four times.

As is apparent from the algorithm, if the number of errors that are actually occurring is less than t, then the two polynomials σ(X), ω(X) are shifted toward high-order locations and output as if they are polynomials of degree 2t after calculations in 2t steps. Therefore, the degrees may be brought into conformity by observing the value of dR_(2t) or the stop condition in the revised Euclidean mutual division method (2) may be changed to dR_(i) <t. FIGS. 3 through 6 show principles of the mutual division unit only, with no consideration given to the time delays of components, etc. For detailed implementation, reference should be made to the article "7R-C601-021(4817) Hardware implementation of a high-speed multiple error-correcting circuit using the Reed-Solomon code (3)".

According to the method B, in view of the fact that it is not necessary for finite-field dividers 103 of the respective mutual division units 101 to operate at the same time, mutual division units 101a, which are the same as the mutual division units 101 except that the finite-field multiplier 103 is removed, are connected in cascade, as shown in FIG. 7 of the accompanying drawings, and R_(i-1) (X), Q_(i-1) (X) from each of the mutual division units 101a are successively selected by a selector switch 105 and divided by a finite-field divider 106, with the quotients fed back to the corresponding mutual division units 101a. The single finite-field divider 106 is thus shared by the mutual division units 101a on a time-division multiplexing basis, resulting in a reduction in the overall gate scale.

However, since the finite-field divider 106 is shared by the mutual division units 101a, a required control circuit is complex in arrangement and its operation speed is not high.

While the finite-field divider 106 is shared, each of the mutual division units 101a requires a circuit for detecting the degrees of polynomials and also detecting when the coefficient of degree dR_(i-1) of R_(i-1) (X) is 0, and also a control circuit such as a data switcher or the like. Consequently, the circuit scale is relatively large.

In the circuit arrangement shown in FIG. 7, it is necessary to input syndromes S_(2t), S_(2t-1), . . ., S₁ which are the coefficients of the syndrome polynomial S(X) successively from the high-order coefficient. Since, however, the syndromes S_(2t), S₂₋₁, . . ., S₁ are simultaneously determined, a conversion circuit for inputting the simultaneously determined syndromes successively from the high-order coefficient is required.

To solve the above problems, a new method referred to as the method C has been proposed in the literature Howard M. Shao, et. al. "A VSLI Design of a Pipeline Reed-Solomon Decoder" IEEE Trans. on Computers Vol. C-34, May 1985.

According to the method C, a circuit for sharing the infinite-field divider 106 and a control circuit therefor are not required because data to be divided can be extracted from one location at all times, unlike the conventional methods. In addition, a circuit for judging operation from the degree of a polynomial and a control circuit are realized by single circuits, respectively. Therefore, the method C can be implemented by a circuit arrangement of reduced circuit scale.

The circuit arrangement which implements the method C is shown in FIG. 8 of the accompanying drawings. As shown in FIG. 8, the circuit arrangement comprises a block (A) 111, a plurality of blocks (B) 112, and a connection switching determination block 113. The block (A) 111 comprises a single unit irrespective of the number t of errors. In the case of a t-symbol error-correcting system, there are required (2t-1) blocks (B) 112.

The block (A) 111 and the clocks (B) 112 include vertical separate groups of registers for storing the coefficients of R_(i) (X), Q_(i) (X), λ_(i) (X), μ_(i) (X) , respectively. The coefficients are stored in the groups of registers successively from the high-order coefficient according to the degree indicated by dR_(i), dQ_(i).

As initial values, the coefficients of R₀ (X)=S(X) are stored successively in registers RR_(2t) -i, . . ., RR₀. Similarly, the coefficients of Q₀ (X)=X^(2t) are stored successively from the high-order coefficient in registers RQ_(2t-1), . . ., RQ₀. Registers Rλ_(2t), . . ., Rλ₀ store 0. With respect to registers Rμ_(2t-1), . . ., Rμ₀, the low-order register Rμ₀ stores 1, and the other registers store 0.

0 is input at all times from the low-order block of the final block (B) 112 to the input terminal.

Whether the coefficient stored in the register RR_(2t) -1 for storing the coefficient of degree dR_(i) of R_(i) (X) in the block (A) 111 is detected by a 0-detecting circuit 115 in the connection switching determination circuit 113.

The connection switching determination circuit 113 has registers DR, DQ for storing dR_(i), dQ_(i), respectively. The registers DR, DQ store 2t-1, 2t as initial values, respectively.

A circuit for setting initial values is required in addition to the circuit arrangement shown in FIG. 8. However, such a circuit is omitted from illustration as it is not essential and is a simple circuit.

Operation of the circuit arrangement shown in FIG. 8 will be described below.

The initial values stored in the registers DR, DQ are compared by a comparator 116. If DR<DQ and the register RR_(2t) -1 ·0 as detected by the 0-detecting circuit 115, the data path switchers as indicated by the dotted lines are operated to select crossed data. Otherwise, the data path switchers select data that are not crossed. All the data path switchers in the circuit arrangement are simultaneously caused to select data.

After the switching of the data path switchers, the coefficient of degree dR_(i-1) of R_(i-1) (X) and the coefficient of degree dQ_(i-1) of Q_(i-1) (X) are input through the data path switcher to a finite-field divider 117. The finite-field divider 117 effects a division E/F on its inputs E, F, and outputs a result S.

Using the result S from the finite-field divider 117, multipliers 119, 120 and adders 121, 122 in the block (A) 111 and the blocks (B) 112 carry out calculations. The coefficients from degree dR_(i) to degree 0 of a polynomial R_(i) (X) are stored in the registers RR_(2t) -1 through RR₀ successively from the high-order coefficient. Similarly, the coefficients from degree dQ_(i) to degree 0 of a polynomial Q_(i) (X) are stored in the registers RQ_(2t) through RQ₀ successively. The coefficients of polynomials λ_(i) (X), μ_(i) (X) are stored in the registers Rλ_(2t) through Rλ₀ and the registers Rμ_(2t-1) through Rμ₀ successively from the high-order coefficient. According to the method C, however, all the coefficients are stored in the registers DR, DQ, RR_(2t-1) through RR₀, RQ_(2t) through RQ₀, Rλ_(2t) through Rλ₀, and Rμ_(2t-1) through Rμ₀. As a result, while all the coefficients of polynomials are serially calculated in one mutual division unit 101 in one step of the revised Euclidean mutual division method (2) according to the method B, each coefficient is calculated in all the calculation units (in the block (A) 111 and the blocks (B) 112) in one step of the revised Euclidean mutual division method (2) according to the method C. Therefore, the method C is greatly simplified though the amount of calculations needed remains unchanged.

As described above with reference to FIG. 8, the coefficient of degree dR_(i) of R_(i) (X) and the coefficient of degree dQ_(i) of Q_(i) (X), which are inputs to the finite-field divider 117, are stored in the registers RR_(2t-1), RQ_(2t-1) in the block (A) 111. Consequently, the inputs to the finite-field divider 117 are always supplied directly behind the data path switcher for R_(i) (X) , Q_(i) (X) in the block (A) 111. Inasmuch as it is possible to input data from the same location at all times to the finite-field divider 117 according to the method B, any additional circuit for sharing the finite-field divider 117 is not required, resulting in a reduction in the overall circuit scale.

According to the method B, the division units 101 are required to have circuits, independent of each other, for determining the magnitudes of dR_(i), dQ_(i) and detecting whether the coefficient of degree dR_(i) of R_(i) (X) is 0 or not to control the respective data path switchers. According to the method C, however, such a process is carried out only by the connection switching determination block 103. Therefore, the circuit scale can be reduced.

Although two multipliers are required in each mutual division unit 101 and hence a total of 4t multipliers are required if the number of correctable symbols is t according to the method B, the number of required multipliers is reduced to 4t-1 according to the method C.

In setting initial values, the coefficients S₁, S₂, . . ., S_(2t) of the input signal R₀ (X), i.e., the syndrome polynomial, are required to be serially input successively from the high-order coefficient according to the method B. According to the method C, however, 2t coefficients of the syndrome polynomial are simultaneously input for initialization.

Syndromes can essentially be determined simultaneously as they are obtained as results of matrix calculations. According to the method B, it is necessary to convert the simultaneously determined syndromes into serial data and input them. According to the method C, since the simultaneously determined coefficients of a syndrome polynomial can be directly input, any circuit for converting them into serial data is not required. Furthermore, a time spent until a result is obtained by the serial data conversion (throughput time) is prevented from increasing.

According to the method C, there are various methods of initializing the registers DR, DQ, RR_(2t-1) through RR₀, RQ_(2t) through RQ₀, Rλ_(2t) through Rλ₀, and Rμ_(2t-1) through Rμ₀. If selectors are connected to the input terminals of the registers DR, DQ, RR_(2t-1) through RR₀, RQ_(2t) through RQ₀, Rλ_(2t) through Rλ₀, and Rμ_(2t-1) through Rμ₀, then it is possible to initialize all the registers at the same time. Alternatively, the registers can be successively initialized by serially inputting initial values of respective polynomials as inputs to low-order registers for respective polynomials.

Now, the execution of only one calculation by one calculation unit in one step according to the revised Euclidean mutual division method (2) will be considered below. That is, the number of calculation units that are essentially required except when two or more calculations are carried out by one calculation unit through multiplexing such as time-division multiplexing will be considered below.

The numbers of calculation units required by the respective methods A, B, C are 8t, 4t, 4t-1, respectively, where t is the number of correctable symbols.

In this case, therefore, ##EQU5##

[STEP 2 ]

    dR.sub.1 -dQ.sub.1 =3-3=0≧0→) normal mode    (22)

In this case, therefore, ##EQU6##

The calculations in these steps are carried out as shown in FIGS. 3 through 6 according to the method B, and as shown in FIGS. 9 through 13 according to the method C.

Multiplications are required in determining R_(i) (X), λ_(i) (X) . With respect to the number of multiplications in each of the steps, the calculation expressed by:

    R.sub.i (X)=Q.sub.0 (X)+(1/α.sup.8)R.sub.0 (X).X     (28)

is carried out to determine R_(i) (X) in step 1. Actually, the coefficients of R₀ (X) and the result α⁷ of the division are multiplied. The polynomial R₀ (X) has four coefficients as it is a polynomial of degree three. Since the coefficient of greatest degree is used for division, it is not required to be calculated. As a result, the following three calculations are carried out:

    α.sup.7 ·α.sup.10, α.sup.7 ·α.sup.5, α.sup.7 ·α.sup.12( 29)

With respect to λ₀ (X), the calculation expressed by:

    λ.sub.1 (X)=μ.sub.0 (X)+(1/α.sup.8)λ.sub.0 (X)·X                                            (30)

is carried out. Since λ₀ (X) is a polynomial of degree 0, only one coefficient is calculated as follows:

    α.sup.7 ·1                                  (31)

Therefore, it can be understood that a total of four calculations are carried out in the step 1.

With respect to the number of calculations in step 2, the following three multiplications are performed to calculate R₂ (X) using the result α⁹ of division:

    α.sup.9 ·α.sup.10, α.sup.9 ·α.sup.5, α.sup.9 ·α.sup.12( 32)

To calculate λ₂ (X), the following one calculation is carried out:

    α.sup.9 ·1                                  (33)

Therefore, a total of four calculations are also carried out in step 2.

In the step 3, two multiplications expressed by:

    α.sup.2 ·α.sup.9, α.sup.2 ·α.sup.6( 34)

are carried out to calculate R₃ (X). In the calculation unit for λ₃ (X), the following two calculations:

    α.sup.2 ·α.sup.7, α.sup.2 ·α.sup.9( 35)

are carried out. Therefore, a total of four calculations are also carried out in step 3.

In the final step 3, two multiplications expressed by:

    α.sup.8 ·α.sup.9, α.sup.8 ·α.sup.6( 36)

are carried out to calculate R₄ (X). In the calculation unit for λ₄ (X), the following two calculations:

    α.sup.8 ·α.sup.7, α.sup.8 ·α.sup.9( 37)

are carried out. Therefore, a total of four calculations are also carried out in step 4.

Consequently, the total number of calculations that are actually required in each of the steps is always 4. This naturally occurs because as the Euclidean mutual division method progresses, the degrees of R_(i) (X) , Q_(i) (X) decrease and the degrees of λ_(i) (X) , μ_(i) (X) increase.

Generally, when R_(i) (X) is calculated in the ith step of the revised Euclidean mutual division method, it is necessary to multiply the coefficients except for the high-order coefficient of R_(i-1) (X) in the cross mode and of Q_(i-1) (X) in the normal mode.

When calculations are carried out, dR_(i-1), dQ_(i) -1 represent the degrees of R_(i-1) (X), Q_(i-1) (X). Since calculations are performed in the cross mode when dR_(i-1) ≧dQ_(i) -1 and in the normal mode when dR_(i-1) ≧dQ_(i-1), the degree of a polynomial required to be multiplied to calculate R_(i) (X) is the smaller one of dR_(i-1), and dQ_(i) -1, i.e., min (dR_(i-1), dQ_(i) -1). At this time, the number of coefficients of the polynomial itself is greater than the degree of the polynominal by 1. The high-order coefficient is not required to be calculated as it is used for division. Consequently, min (dR_(i-1), dQ_(i) -1) calculations are required to calculate R_(i) (X) .

Calculations for λ_(i) (X) are performed by λ_(i-1) (X) , μ_(i-1) (X) . To calculate λ_(i) (X) , all coefficients of λ_(i-1) (X) are required to be multiplied in the cross mode, and all coefficients of μ_(i-1) (X) are required to be multiplied in the normal mode.

In the normal and cross modes in which calculations are actually carried out other than the shift mode, when the following conditions are satisfied:

    dλ.sub.i +dQ.sub.i =b 2t

    dμ.sub.i <dλ.sub.i

    dR.sub.i-1 <dQ.sub.i-1                                     ( 38)

where dλ_(i) is the degree of λ_(i) (X) and dμ_(i) is the degree of μ_(i) (X) in the calculations in the ith step, the calculations are always carried out in the cross mode, and the coefficients other than the high-order coefficient (of degree dR_(i-1)) of a polynomial R_(i-1) (X) of degree dR_(i-1) are multiplied for the calculation of R_(i) (X). Accordingly, dR_(i-1) calculations are necessary. For the calculation of λ_(i) (X), all (dλ_(i-1) +1) coefficients of λ_(i-1) (X) are required to be multiplied.

Therefore, the number N_(cross) of all multiplications in the cross mode is expressed by: ##EQU7## As a consequence, the number N_(cross) of entire multiplications in the cross mode is 2t or smaller.

In the normal mode when dR_(i-1) ≦dQ_(i) -1, the coefficients other than the high-order coefficient (of degree dQ_(i) -1) of a polynomial Q_(i-1) (X) of degree dQ_(i-1) are required to be multiplied for the calculation of R_(i) (X) , and dQ_(i-1) multiplications are needed. Likewise, to calculate λ_(i) (X), all (dR_(i-1) +1) coefficients of μ_(i-1) (X) are required to be multiplied.

Therefore, the number N_(normal) of all multiplications in the normal mode is expressed by: ##EQU8## As a consequence, the number N_(cross) of entire multiplications in the normal mode is also 2t or smaller.

As described above, the total number of multiplications in each step required by the algorithm itself is always 2t or less. Thus, the number of calculation units that are essentially needed is only 2t.

In the above example, the number of calculation units should therefore be 4. Actual systems which implement the methods A, B, C, however, require a number of calculation units that are more than twice 2t, and hence are of a very wasteful circuit arrangement.

The number of coefficients that need to be held for calculations will be considered below.

dR_(i), dQ_(i), dλ_(i), dμ_(i) represent the respective degrees of R_(i) (X), Q_(i) (X), λ_(i) (X), μ_(i) (X) in actual calculation steps in the normal and cross modes. These degrees have the following relationship:

    dR.sub.i +dμ.sub.i <dQ.sub.i +dλ.sub.i =2t       (41)

As can be seen from the above, the sum of the degrees of Q_(i) (X), λ_(i) (X) is 2t at all times, and the sum of the degrees of R_(i) (X), μ_(i) (X) is 2t-1 or less.

Therefore, the number of registers required to store the coefficients of Q_(i) (X), λ_(i) (X) may be 2t+2, and the number of registers required to store the coefficients of R_(i) (X), μ_(i) (X) may be 2t+1. The total number of registers required may thus be 4t+3.

However, the system which implements the method C employs (8t+1) registers and hence is a highly wasteful arrangement.

As described above, in either one of the methods A, B, C, the number of multiplications per step and the number of registers are more than twice the numbers that are required by the algorithm. The systems which implement the methods A, B, C are of a highly wasteful circuit arrangement.

The calculations in the above example will be described once more for an easier understanding. According to the method C, the calculations carried out as shown in FIGS. 9 through 13 are indicated by the following tables:

                                      TABLE 1-1                                    __________________________________________________________________________     [STEP 1]                                                                       Cross mode dR = 3, dQ = 4                                                            Calculation                                                                             Calculation                                                                              Calculation                                                                            Calculation                                   R.sub.i                                                                           Q.sub.i                                                                           of R     of Q  λ.sub.i                                                                   μ.sub.i                                                                       of λ                                                                            of μ                                       __________________________________________________________________________     α.sup.8                                                                     1  S = Q/R = α.sup.7                                                                 --    0 0 λ = μ                                                                        λ = μ                               α.sup.10                                                                    0  R = Q + α.sup.7 × R                                                         Q = R 0 0 λ = μ                                                                        λ = μ                               α.sup.5                                                                     0  R = Q + α.sup.7 × R                                                         Q = R 0 0 λ = μ                                                                        λ = μ                               α.sup.12                                                                    0  R = Q + α.sup.7 × R                                                         Q = R 1 0 λ = μ  + α.sup.7 ×                                       λ                                                                               λ = μ                               __________________________________________________________________________

                                      TABLE 1-2                                    __________________________________________________________________________     [STEP 2]                                                                       Normal mode dR = 3, dQ = 3                                                           Calculation                                                                             Calculation                                                                              Calculation                                                                             Calculation                                  R.sub.i                                                                           Q.sub.i                                                                           of R     of Q  λ.sub.i                                                                   μ.sub.i                                                                       of λ                                                                             of μ                                      __________________________________________________________________________     α.sup.2                                                                     α.sup.8                                                                     S = R/Q = α.sup.9                                                                 --    0 0 λ = λ                                                                     μ = μ                                  α.sup.12                                                                    α.sup.10                                                                    R = R + α.sup.9 × R                                                         Q = Q 0 0 λ = λ                                                                     μ = μ                                  α.sup.4                                                                     α.sup.5                                                                     R = R + α.sup.9 × Q                                                         Q = Q α.sup.7                                                                    0 λ = λ                                                                     μ = μ                                  0  α.sup.12                                                                    R = R + α.sup.9 × Q                                                         Q = Q 1 1 λ = μ + α.sup.9                                                         μ = μ.                                 __________________________________________________________________________

                                      TABLE 1-3                                    __________________________________________________________________________     [STEP 3]                                                                       Cross mode dR = 2, dQ = 3                                                           Calculation                                                                             Calculation                                                                              Calculation                                                                             Calculation                                   R.sub.i                                                                          Q.sub.i                                                                           of R     of Q  λ.sub.i                                                                   μ.sub.i                                                                       of λ                                                                             of μ                                       __________________________________________________________________________     α.sup.6                                                                    α.sup.8                                                                     S = Q/R = α.sup.2                                                                 --    0 0 λ = μ                                                                         μ = λ                               α.sup.9                                                                    α.sup.10                                                                    R = Q + α.sup.2 × R                                                         Q = R α.sup.7                                                                    0 λ = μ + α.sup.2                                                         μ = λ.                              α.sup.6                                                                    α.sup.5                                                                     R = Q + α.sup.2 × R                                                         Q = R α.sup.9                                                                    0 λ = μ + α.sup.2                                                         μ = λ.                              0 α.sup.12                                                                    R = Q    Q = R 0 1 λ = μ                                                                         μ = λ                               __________________________________________________________________________

                                      TABLE 1-4                                    __________________________________________________________________________     [STEP 4]                                                                       Normal mode dR = 2, dQ = 2                                                          Calculation                                                                             Calculation                                                                               Calculation                                                                             Calculation                                  R.sub.i                                                                           Q.sub.i                                                                          of R     of Q  λ.sub.i                                                                    μ.sub.i                                                                       of λ                                                                             of μ                                      __________________________________________________________________________     α.sup.14                                                                    α.sup.6                                                                    S = R/Q = α.sup.8                                                                 --    α.sup.9                                                                     0 λ = λ                                                                     μ = μ                                  α.sup.4                                                                     α.sup.9                                                                    R = R + α.sup.8 × Q                                                         Q = Q α.sup.11                                                                    α.sup.7                                                                    λ = λ + α.sup.8 ×                                             μ = μ                                  α.sup.12                                                                    α.sup.6                                                                    R = R + α.sup.8 × Q                                                         Q = Q 1  α.sup.9                                                                    λ = λ + α.sup.8 ×                                             μ  = μ                                 0  0 R = R    Q = Q 0  0 λ = λ                                                                     μ = μ                                  __________________________________________________________________________

As is apparent from the above tables, according to the method C, only four multiplications are carried out in each step, and the remaining calculation units are not in operation. The locations where λ_(i) (X), μ_(i) (X) are stored are apparently shifted toward high-order locations as the calculations progress, and hence the locations where the calculation units are used vary in each step. Therefore, the number of calculation units cannot be reduced according to the method C.

OBJECTS AND SUMMARY OF THE INVENTION

It is an object of the present invention to provide an Euclidean mutual division circuit having (2t+1) calculation units, the number of which is greater, by 1, than the minimum number 2t of calculations per step of the Euclidean mutual division method, and a minimum number of (4t+3) registers, for accomplishing an algorithm thereby to greatly reduce a circuit scale and to operate at high speed for an increased throughput.

Another object of the present invention is to provide an Euclidean mutual division circuit having fewer calculation units than (2t+1) which are used in a time-division multiplexing process for carrying out (2t+1) calculations per step which are required of the Euclidean mutual division method.

According to the present invention, there is provided an Euclidean mutual division circuit comprising a plurality of calculation devices each composed of two registers for storing R_(i) (X) , Q_(i) (X), λ_(i) (X), μ_(i) (X) and a calculation unit, the calculation devices being cascaded in a number corresponding to one more than the minimum number of calculations per step of a Euclidean mutual division for receiving a result of division and a value output from a preceding stage, the calculation devices being responsive to a switching command for effecting either a normal connection calculation or a cross-connection calculation, a division unit having two registers for storing R_(i) (X), Q_(i) (X), λ_(i) (X), μ_(i) (X) and a divider, for receiving a value output from a final stage of the calculation devices, for effecting either a normal-connection division or a cross-connection division in response to the switching command, and supplying the result of division to each of the calculation devices, and a control unit for generating a switching command indicative of either a normal connection or a cross connection based on a predetermined initial value and a value stored in a register in the division unit, and supplying the generated switching command to the calculation devices and the division unit.

The control unit generates a switching command for indicating a normal connection or a cross connection based on a comparison of a predetermined initial value and a value stored in a register in the division unit. At the same time, the calculation devices receive a result of division and a value output from a preceding stage and is responsive to a switching command for effecting either a normal connection calculation or a cross-connection calculation. The division unit receives a value output from a final stage of the calculation devices, effects either a normal-connection division or a cross-connection division in response to the switching command, and supplies the result of division to each of the calculation devices for effecting Euclidean mutual division.

According to the present invention, there is also provided an Euclidean mutual division circuit comprising a block (heretofore "MLT block") having a group of A-side registers divided depending on a degree of multiplexing for storing coefficients of R_(i) (X), Q_(i) (X), a group of B-side registers divided depending on the degree of multiplexing for storing coefficients of λ_(i) (X), μ_(i) (X) , and a number of calculation units as great as (2t+1) divided by the degree of multiplexing, the MLT block being capable of receiving a result of division and values output from the A- and B-side registers, and responsive to a switching command for effecting either a normal connection calculation or a cross-connection calculation or a shift-connection calculation in each clock cycle in each step, a division unit having a register for storing R_(i) (X), Q_(i) (X), a register for storing λ_(i) (X), μ_(i) (X), and a divider for dividing coefficients stored in the registers, for receiving a value output from a final stage of the MLT block, effecting either a normal-connection division or a cross-connection division in response to the switching command, and supplying the result of division to each of the calculation units until each step is finished, and a determination and control unit for generating a switching command indicative of either a normal connection or a cross connection or a shift connection based on a comparison of a predetermined value and a value stored in a register in the division unit, and supplying the generated switching command to the MLT block and the division unit.

The determination and control unit generates a switching command for indicating a normal connection or a cross connection or a shift connection based on a predetermined value set in each step and a value stored in a register in the division unit. The division unit receives a value output from a final stage of the MLT block, and is responsive to a switching command for effecting either a normal connection calculation or a cross-connection calculation. The result of the calculation is supplied to each of the calculation units until each step is finished. Concurrent with this, in each clock cycle in each step, each of the calculation units of the MLT block receives a result of division and values output from the A- and B-side registers, and is responsive to a switching command for effecting either a normal connection calculation or a cross-connection calculation or a shift-connection calculation, for carrying out Euclidean mutual divisions.

The above and other objects, features, and advantages of the present invention will become apparent from the following description of illustrative embodiments thereof to be read in conjunction with the accompanying drawings, in which like reference numerals represent the same or similar objects.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a mutual division unit for use in a conventional Euclidean mutual division circuit;

FIG. 2 is a block diagram of a conventional Euclidean mutual division circuit;

FIG. 3 is a block diagram showing the manner in which the Euclidean mutual division circuit shown in FIG. 2 operates;

FIGS. 4-6 are block diagrams showing the manner in which the Euclidean mutual division circuit shown in FIG. 2 operates;

FIG. 7 is a block diagram of another conventional Euclidean mutual division circuit;

FIG. 8 is a block diagram of still another conventional Euclidean mutual division circuit;

FIGS. 9-13 are block diagrams showing the manner in which the Euclidean mutual division circuit shown in FIG. 8 operates;

FIG. 14 is a block diagram of an Euclidean mutual division circuit according to a first embodiment of the present invention;

FIG. 15 is a detailed circuit diagram of each calculation unit in the Euclidean mutual division circuit shown in FIG. 14;

FIGS. 16-20 are block diagrams showing the manner in which the Euclidean mutual division circuit shown in FIG. 14 operates;

FIG. 21 is a block diagram of an Euclidean mutual division circuit according to a second embodiment of the present invention;

FIG. 22 is a detailed circuit diagram of each calculation unit in the Euclidean mutual division circuit shown in FIG. 21;

FIG. 23-35 are block diagrams showing the manner in which the Euclidean mutual division circuit shown in FIG. 21 operates.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

A method (hereinafter referred to as a "method D") according to a first embodiment of the present invention is proposed which can perform an algorithm with (2t+1) calculation units, the number of which is greater, by 1, than the minimum number 2t of calculations per step of the Euclidean mutual division method, and a minimum number of (4t+3) registers.

The method D is of great advantage because it can accomplish the Euclidean mutual division algorithm with calculation units much fewer than those required by the conventionally proposed methods A, B, and C.

An Euclidean mutual division circuit for carrying out the method D is schematically shown in FIG. 14. As shown in FIG. 14, the Euclidean mutual division circuit basically comprises a DIV block 131 and an MLT block 132.

The DIV block 131 comprises a single division unit 133 for a high-order coefficient, the division unit 133 having a finite-field divider 146, and a control unit 134 for detecting the magnitudes of dR_(i), dQ_(i) and also detecting whether the coefficient of degree dR_(i) of R_(i) (X) is equal to zero to generate a signal for controlling data path switchers in the DIV block 131 and the MLT block 132.

If calculation units are not used in a multiplexing fashion for effecting calculations on a plurality of coefficients, then the MLT block 132 requires (2t+1) calculation units 135₁ through 135_(2t+1) when it is implemented as a system capable of error-correcting t symbols.

As shown in FIG. 15, each of the calculation units 135₁ through 135_(2t+1) comprises three data path switchers 136, 137, 138, a multiplier 139, and an adder 140, for calculating a given finite field.

The DIV block 131 and the MLT block 132 have a lefthand group of registers 141₀ through 141_(2t), which are referred to as A-side registers, and a righthand group of registers 142₀ through 142_(2t+1), which are referred to as B-side registers. The A-side registers 141₀ through 141_(2t) in the DIV block 131 and the MLT block 132 are reduced to (2t+1) registers, and the B-side registers 142₀ through 142_(2t+1) in the DIV block 131 and the MLT block 132 are reduced to (2t+2) registers, the total number of the registers being thus reduced to 4t+3. The number of calculation units 135₁ through 135_(2t+1) is reduced to 2t+1.

The DIV block 131 has a register 143 for DR and a register 144 for DQ, the registers 143, 144 being indicative of the degrees of coefficients stored in the register 141₀ for A₀ and the register 142₀ for B₀. When actual calculations are carried out in cross and normal modes (described later on), the registers 143, 144 indicate the degrees of R_(i) (X), Q_(i) (X).

The control unit 134 in the DIV block 131 has an operation mode determination and control circuit 145 for determining an operation mode from the results of comparison of DR, DQ stored in the registers 143, 144 and the result of detection as to whether A₀ is 0 from the register 141₀. Based on the determined operation mode and the value of DR, the calculation units 135₁ through 135_(2t+1) in the MLT block 132 are controlled independently of each other. No A-side register is required on the input side of the final calculation unit 135_(2t+1). The circuit shown in FIG. 19 is conceptual, and the A-side register which is shown on the input side of each of the calculation units 135₁ through 135_(2t) is omitted from the final calculation unit 135_(2t+1). The final calculation unit 135_(2t+1) may be made simpler in circuit arrangement as the input applied to the A-side thereof is always 0.

A process of storing coefficients of each polynomial to be calculated will be described below.

According to the method D, storage locations for the coefficients of Q_(i) (X), λ_(i) (X) and storage locations for the coefficients of R_(i) (X), μ_(i) (X) are used respectively in common.

In the calculation in each step of the Euclidean mutual division method, the relationship expressed by:

    dR.sub.i +dμi<dQ.sub.i +dλ.sub.i =2t             (42)

is satisfied. This relationship indicates that the sum of the degrees of Q_(i) (X), λ_(i) (X) is always 2t. Therefore, even if the degree of λ_(i) (X) increases as the degree of Q_(i) (X) decreases, the sum of the numbers of registers for storing the coefficients of polynomials is 2t+2 at all times. Similarly, since the sum of the degrees of R_(i) (X), μ_(i) (X) is 2t-1 or smaller, the sum of the numbers of registers for storing the coefficients may be at most 2t+1.

The registers 141₀ through 1412t shown in FIG. 14 are registers for storing the coefficients of R_(i) (X) and μ_(i) (X).

In each calculation step, the coefficients of the polynomial R_(i) (X) of degree dR_(i) are stored in the A-side registers 141₀ through 141_(2t) successively from the high-order coefficient (of degree dR_(i)).

Following the coefficient of degree 0 of R_(i) (X), the coefficients of μ_(i) (X) are stored successively from the low-order coefficient (of degree 0). Therefore, the coefficients of R_(i) (X) are stored successively from the high-order coefficient in the A-side registers 141₀ through 141_(2t) successively from the high-order register 141₀, and then the coefficients of μ_(i) (X) are stored successively from the low-order coefficients in the A-side registers 141₀ through 141_(2t).

The registers 142₀ through 142_(2t+1) shown in FIG. 14 are registers for storing the coefficients of Q_(i) (X), λ_(i) (X).

In each calculation step, the coefficients of the polynomial Q_(i) (X) of degree dQ_(i) are stored in the registers 142₀ through 142_(2t+1) successively from the high-order coefficient (of degree dQ_(i)). Following the coefficient of degree 0 of Q_(i) (X), the coefficients of λ_(i) (X) are stored successively from the low-order coefficient (of degree 0).

Inasmuch as dQ_(i) +dλ_(i) =2t and the sum of the degrees is constant at all times, the coefficients of Q_(i) (X) are stored in the B-side registers 142₀ through 142_(2t+1) successively from the high-order register 142₀, and then the coefficients of λ_(i) (X) are stored in the B-side registers 142₀ through 142_(2t+1) successively from the low-order register 142_(2t+1).

As described above, as the coefficient-storing registers 141₀ through 141_(2t), 142₀ through 142_(2t+1) are used in common, the entire circuit may be composed of (4t+3) registers.

In setting initial values, initial values for dR₀, dQ₀, dλ₀, dμ₀ which indicate the respective degrees of R₀ (X), Q₀ (X), λ₀ (X), μ₀ (X) are considered to be 2t-1, 2t, 0, -1, respectively. The initial value for the degree dμ₀ of μ₀ (X) is selected to be -1 for the convenience of the algorithm.

Therefore, 2t coefficients (syndromes) of R₀ (X)=S(X) are stored in the registers 141₀ through 141_(2t+1). Since the degree of μ₀ (X) is -1 for the convenience of the algorithm, the number of coefficients of μ₀ (X) is 0, i.e., there is no corresponding register, so that the register 141_(2t) is set to 0.

As regards initial values for the registers 142₀ through 142_(2t), since Q₀ (X)=X_(2t),1 is stored in the register 142₀, and 0 is stored in the registers 142₁ through 142_(2t). As λ₀ (X)=1, 1 is stored in the register 141_(2t+1).

The initial values 2t -1, 2t are stored in the registers 143, 144 which store DR and DQ, respectively. A circuit for setting initial values is required in addition to the circuit arrangement shown in FIG. 14. However, such a circuit is omitted from illustration as it is not essential and is a simple circuit.

Operation of the circuit arrangement shown in FIG. 14 will be described below.

After the registers have been set to initial values, the Euclidean mutual division method is carried out one step per clock pulse. The registers 141₀, 141₁, 141₂, . . ., 142₀, . . ., 142_(2t+1) hold results of calculations in each clock cycle.

Calculations in an ith step will be considered below. The values of DR_(i-1), DQ_(i-1) stored in the registers 143, 144 are compared by the operation mode determination and control circuit 145. If DR_(i-1) <DQ_(i-1) and it is indicated by a 0-detecting circuit 147 that the register 141₀ which stores the high-order coefficient of R_(i-1) (X) holds 0, then the operation mode determination and control circuit 145 recognizes the cross mode. Otherwise, the operation mode determination and control circuit 145 recognizes the normal mode.

When the operation mode is determined, data path switchers 148 and 149 in the DIV block 131 switches to select crossed data or normal data depending on the determined operation mode. The value of the register 141₀, i.e., the high-order coefficient (of degree dR_(i-1)) of R_(i-1) (X), and the value of the register 142₀, i.e., the high-order coefficient (of degree dQ_(i-1)) of Q_(i-1) (X) pass through the data path switcher 148 to the finite-field divider 146. The finite-field divider 146 effects a division F/G on its inputs F and G, and outputs a result S.

Each of the calculation units 135₁ through 135_(2t+1) in the MLT block 132 carries out calculations depending on the operation mode determined by the DIV block 131. A calculation unit having a number j effects calculations according to the following tables, with respect to a register having a register number (suffix) j among the registers 141₁ through 141_(2t), 142₁ through 142_(2t+1) which store coefficients to be calculated:

                                      TABLE 2-1                                    __________________________________________________________________________     Calculation table in the cross mode                                            Cross mode                                                                     Calculation  Calculation                                                       unit number  mode  Calculations                                                __________________________________________________________________________     0 < j ≦ dR.sub.i-1                                                                   1     A' = A × S + B                                                                      B' = A                                           dR.sub.i-1 < j ≦ dQ.sub.i-1                                                          2     A' = B     B' = A                                           dQ.sub.i-1 < j                                                                              3     A' = B     B' = B × S + A                             __________________________________________________________________________

                                      TABLE 2-2                                    __________________________________________________________________________     Calculation table in the normal mode                                           Normal mode                                                                    Calculation                                                                               Calculation                                                         unit number                                                                               mode    Calculations                                                __________________________________________________________________________     0 < j ≦ dQ.sub.i-1                                                                 4       A' = B × S + A                                                                    B' = B                                             dQ.sub.i-1 < j ≦ dR.sub.i-1                                                        5       A' = A   B' = B                                             dR.sub.i-1 < j                                                                            6       A' = A   B' = A × S + B                               __________________________________________________________________________

                  TABLE 2-3                                                        ______________________________________                                         Calculation table in the shift mode                                            Shift mode                                                                     Calculation                                                                               Calculation                                                         unit number                                                                               mode          Calculations                                          ______________________________________                                         All j      5             A' = A   B' = B                                       ______________________________________                                    

The calculated A' and B' values are stored in the next registers, completing calculations in one step of the revised Euclidean mutual division method. The above calculating process is repeated 2t times to determine an error locator polynomial σ(X) and an error evaluator polynomial ω(X).

Usually, according to the algorithm of the revised Euclidean mutual division method, the same calculations as those effected on the set of the polynomials R_(i-1) (X), Q_(i-1) (X) are effected on the set of the polynomials λ_(i-1) (X), μ_(i-1) (X) . According to the method D, the polynomials R_(i-1) (X), Q_(i-1) (X) are stored respectively in the A-side registers 141₀ through 141_(2t), and the B-side registers 142₀ through 142_(2t+1), and the polynomials λ_(i-1) (X), μ_(i-1) (X) are stored inversely therein.

To carry out the same calculations on R_(i-1) (X), λ_(i-1) (X) , the coefficients of λ_(i-1) (X) which are stored inversely may be inverted, calculated, and the result may be stored inversely again.

Therefore, the polynomials R_(i-1) (X), Q_(i-1) (X) may be calculated a number of times corresponding to the number of coefficients of the polynomial whose degree is smaller in either the cross mode or the normal mode. No calculations are needed for those registers which store coefficients that are not next to each other of R_(i-1) (X), Q_(i-1) (X). This also holds true for λ_(i-1) (X), μ_(i-1) (X).

For example, in the cross mode, since dR_(i-1) <dQ_(i-1), the polynomials R_(i-1) (X), Q_(i-1) (X) are calculated only when the register number j is in the range of 0<j<dR_(i-1).

When the register number j is in the range of dR_(i-1) <j ≦dQ_(i-1), the coefficients of R_(i-1) (X) are stored in the A-side registers 141₀ through 141_(2t), and the coefficients of λ_(i-1) (X) are stored in the B-side registers 142₀ through 142_(2t+1). In this case, no calculations are necessary on the coefficients stored in the registers, but the stored coefficients may only be switched around.

Of the registers 141₀ through 141_(2t), 142₀ through 142_(2t+1), those A-side registers whose register number is in the range of dQ_(i-1) <j store μ_(i-1) (X) and those B-side registers whose register number is in the same range store λ_(i-1) (X). The values in the A- and B-side registers may be inverted, and the same calculations as those on dR_(i-1) (X), dQ_(i-1) (X) may be carried out thereon, and the results may be stored inversely again.

Consequently, the number of calculations per step is 2t at all times, and the calculations can be performed by the (2t+1) calculation units according to the method D.

While no calculations need to be carried out in the shift mode, the shift mode may be effected in the same manner as the normal mode in which the output S from the finite-field divider 146 is 0.

For example, it is possible to effect calculations according to the following tables:

                                      TABLE 3-1                                    __________________________________________________________________________     Calculation table in the cross mode                                            Cross mode                                                                                           Calculation                                              Calculation unit number                                                                              mode  Calculations                                       __________________________________________________________________________     0 < j ≦ min (dR.sub.i-1, dQ.sub.i-1)                                                          1     A' = A × S + B                                                                    B' = A                                    min (dR.sub.i-1, dQ.sub.i-1) < j ≦ max (dR.sub.i-1,                                           2Q.sub.i-1)                                                                          A' = B   B' = A                                    max (dR.sub.i-1, dQ.sub.i-1) < j                                                                     3     A' = B   B' = B × S + A                      __________________________________________________________________________

                                      TABLE 3-2                                    __________________________________________________________________________     Calculation table in the normal and shift modes                                Normal mode                                                                                          Calculation                                              Calculation unit number                                                                              mode  Calculations                                       __________________________________________________________________________     0 < j ≦ min (dR.sub.i-1, dQ.sub.i-1)                                                          4     A' = B × S + A                                                                    B' = B                                    min (dR.sub.i-1, dQ.sub.i-1) < j ≦ max (dR.sub.i-1,                                           5Q.sub.i-1)                                                                          A' = A   B' = B                                    max (dR.sub.i-1, dQ.sub.i-1) < j                                                                     6     A' = A   B' = A × S + B                      __________________________________________________________________________

The calculation units 135₁ through 135_(2t+1) may be of any specific arrangement insofar as it can perform at least the six calculations indicated in the tables 3-1, 3-2 above.

For example, each of the calculation units 135₁ through 135_(2t+1) may be of a circuit arrangement as shown in FIG. 15. The circuit shown in FIG. 15 comprises three data path switchers 136, 137, 138, a multiplier 139, and an adder 140. The necessary calculations can be accomplished by controlling the data path switchers 136, 137, 138 according to the table given below. The same calculations may also be carried out by a control process other than the switcher control process in the table.

                                      TABLE 4                                      __________________________________________________________________________     Control of the data path switches                                              Calculation             Switcher                                                                            Switcher                                                                            Switcher                                     mode  Calculations      137  138  136                                          __________________________________________________________________________     1     A' = A × S + B                                                                    B' = A   cross                                                                               through                                                                             normal                                       2     A' = B   B' = A   cross                                                                               clear                                                                               normal                                       3     A' = B   B' = B × S + A                                                                    normal                                                                              through                                                                             cross                                        4     A' = B × S + A                                                                    B' = B   normal                                                                              through                                                                             normal                                       5     A' = A   B' = B   normal                                                                              clear                                                                               normal                                       6     A' = A   B' = A × S + B                                                                    cross                                                                               through                                                                             cross                                        __________________________________________________________________________

In the above table, "normal" for the data path switchers 137, 136 indicates switching to terminals I--I', and "cross" for the data path switchers 137, 136 indicates switching to terminals H--H' and "through" for the data path switcher 138 indicates switching to a terminal I, and "clear" for the data path switcher 138 indicates switching to a terminal H to input O.

Operation of the Euclidean mutual division circuit shown in FIG. 14, which is composed of the calculation units 135₁ through 135_(2t+1) shown in FIG. 15, is shown by way of example in FIGS. 16 through 20. The illustrated example is the same as the example described above in which t=2 symbols are error-corrected using a finite field GF(2⁴) that is defined using an irreducible polynomial g(X)=X⁴ +X+1. It is assumed that all the registers 141₀ through 141₄, 142₀ through 142₅, 143 and 144 have been initialized.

In a first step shown in FIG. 16, since the value (dR₀) of DR stored in the register 143 is 3, and the value (dQ₀) of DQ stored in the register 144 is 4, the condition DR<DQ is satisfied, and the coefficient of degree dR₀ of R₀ (X) stored in the register 141₀ is α⁸ which is nonzero. Therefore, the operation mode is determined as the cross mode. The data path switcher 148 on the input side of the finite-field divider 146 is switched into the cross state, and the finite-field divider 146 carries out a calculation 1/α⁸. The result S=α⁷ of division is output to the calculation units 135₁ through 135₅.

In FIG. 16, therefore, the cross mode of operation is carried out.

Consequently, the calculation units 135₁ through 135₅ in the MLT block 132 carries out calculations based on the calculation table 2-1 in the cross mode.

According to the table 2-1, the data stored in the registers 141_(j), 142_(j) whose register number is j are calculated in one of three different calculation modes depending on whether the register number j is in the range of 0<j≦dR_(i-1) dR_(i-1) <j ≦dQ_(i-1), or dQ_(i-1) <j.

For example, in the first step shown in FIG. 16, since dR₀ =3, dQ₀ =4, the calculation units 135₁ through 135₃ corresponding to respective register numbers 1 through 3 carry out calculations indicated by A'=A×S+B and B'=A. For such calculations, the data path switcher 137 is in the cross state, the data path switcher 138 is in the through state, and the data path switcher 136 is in the normal state.

The calculation unit 135₄ corresponding to a register number 4 carries out calculations indicated by A'=B and B'=A. For such calculations, the data path switcher 137 is in the cross state, the data path switcher 138 in the clear state, and the data path switcher 136 in the normal state.

The calculation units 135₅, 135₆ corresponding to respective register numbers 5, 6 carry out calculations indicated by A'=B and B'=B×S+A. For such calculations, the data path switcher 137 is in the normal state, the data path switcher 138 is in the through state, and the data path switcher 136 is in the cross state.

In the first step, the data path switchers 136, 137, 138 in the calculation units 135₁ through 135₅ operate as shown in FIG. 16. The calculated results are stored in the registers 141₀ through 141₄, 142₀ through 142₅ in the next clock cycle as shown in FIG. 17.

A comparison between FIG. 10 showing the results according to method C and FIG. 17 showing the results according to method D indicates that while the coefficients of λ₁ (X), μ₁ (X) are stored behind R_(i) (X), Q_(i) (X) successively from the high-order coefficient in FIG. 10, the coefficients of λ₁ (X), μ₁ (X) are stored in inverted locations, i.e., successively from the low-order coefficient in FIG. 17. In FIG. 17, therefore, since the coefficients whose value is 0 that are essentially unnecessary are not required to be stored, no wasteful calculations are carried out, and as a result, the number of calculation units is greatly reduced.

In a second step shown in FIG. 17, since the value (dR_(i)) of DR stored in the register 143 is 3, and the value (dQ_(i)) of DQ stored in the register 144 is 3, the condition DR≧DQ is satisfied. Therefore, the operation mode is determined as the normal mode. The data path switcher 148 on the input side of the finite-field divider 146 is switched into the normal state, and the finite-field divider 146 carries out a calculation α² /α⁸. The result S=α⁹ of division is output to the calculation units 135₁ through 135₅.

In FIG. 17, therefore, the cross mode of operation is carried out. Consequently, the calculation units 135₁ through 135₅ in the MLT block 132 carries out calculations based on the calculation table 2-2 in the normal mode.

In the second step shown in FIG. 17, the calculation units 135₁ through 135₃ corresponding to respective register numbers 1 through 3 carry out calculations indicated by A'=B×S+A and B'=B. For such calculations, the data path switcher 137 is in the normal state, the data path switcher 138 is in the through state, and the data path switcher 136 is in the normal state.

The calculation units 135₄, 135₅ corresponding to the registers 141₄, 142₄, 142₅ having register numbers 4, 5 carry out calculations indicated by A'=A and B'=A×S+ B. For such calculations, the data path switcher 137 is in the cross state, the data path switcher 138 is in the through state, and the data path switcher 136 is in the cross state.

In the second step, the data path switchers 136, 137, 138 in the calculation units 135₁ through 135₅ operate as shown in FIG. 17. The calculated results are stored in the registers 141₀ through 141₄, 142₀ through 142₅ in the next clock cycle as shown in FIG. 18.

The first and third steps are effected in the cross mode, and the second and fourth steps are effected in the normal mode as shown in FIGS. 16 through 19, until finally the polynomials σ(X), ω(X) are obtained as shown in FIG. 20. It should be noted that the locations and sequence of the obtained coefficients of σ(X) are inverted relationship to those of the coefficients which are obtained according to method C as shown in FIG. 13.

The calculations corresponding to the operations shown in FIGS. 16 through 20, respectively, are summarized in the following tables:

    ______________________________________                                         [STEP 1]                                                                       Cross mode dR = 3, dQ = 4                                                                            Calculations                                                                               Calculations                                 i      A.sub.i B.sub.i                                                                               on A side   on B side                                    ______________________________________                                         0      α.sup.8                                                                          1      S = B/A = α.sup.7                                                                    --                                           1      α.sup.10                                                                         0      A = B + α.sup.7 × A                                                            B = A                                        2      α.sup.5                                                                          0      A = B + α.sup.7 × A                                                            B = A                                        3      α.sup.12                                                                         0      A = B + α.sup.7 × A                                                            B = A                                        4      0       0      A = B       B = A                                        5      0       1      A = B       B = A + α.sup.7                        ______________________________________                                                                           × B                               

    ______________________________________                                         [STEP 2]                                                                       Normal mode dR = 3, dQ = 3                                                                          Calculations                                                                               Calculations                                  i    A.sub.i B.sub.i on A side   on B side                                     ______________________________________                                         0    α.sup.2                                                                          α.sup.8                                                                          S = A/B = α.sup.9                                                                    --                                            1    α.sup.12                                                                         α.sup.10                                                                         A = A + α.sup.9 × B                                                            B = B                                         2    α.sup.4                                                                          α.sup.5                                                                          A = A + α.sup.9 × B                                                            B = B                                         3    0       α.sup.12                                                                         A = A + α.sup.9 × B                                                            B = B                                         4    1       0       A = A       B = B + α.sup.9 × A               5    0       α.sup.7                                                                          A = A       B = B + α.sup.9 × A               ______________________________________                                    

    ______________________________________                                         [STEP 3]                                                                       Cross mode dR = 2, dQ = 3                                                                           Calculations                                                                               Calculations                                  i    A.sub.i B.sub.i on A side   on B side                                     ______________________________________                                         0    α.sup.6                                                                          α.sup.8                                                                          S = B/A = α.sup.2                                                                    --                                            1    α.sup.9                                                                          α.sup.10                                                                         A = B + α.sup.2 × A                                                            B = A                                         2    α.sup.6                                                                          α.sup.5                                                                          A = B + α.sup.2 × A                                                            B = A                                         3    1       α.sup.12                                                                         A = B       B = A                                         4    0       α.sup.9                                                                          A = B       B = A + α.sup.2 × B               5    0       α.sup.7                                                                          A = B       B = A + α.sup.2 × B               ______________________________________                                    

    ______________________________________                                         [STEP 4]                                                                       Normal mode dR = 2, dQ = 2                                                                          Calculations                                                                               Calculations                                  i    A.sub.i B.sub.i on A side   on B side                                     ______________________________________                                         0    α.sup.14                                                                         α.sup.6                                                                          S = A/B = α.sup.8                                                                    --                                            1    α.sup.4                                                                          α.sup.9                                                                          A = A + α.sup.8 × B                                                            B = B                                         2    α.sup.12                                                                         α.sup.6                                                                          A = A + α.sup.8 × B                                                            B = B                                         3    α.sup.9                                                                          1       A = A       B = B + α.sup.8 × A               4    α.sup.7                                                                          α.sup.11                                                                         A = A       B = B + α.sup.8 × A               5    0       α.sup.9                                                                          A = A       B = B + α.sup.8 × A               ______________________________________                                    

As described above, although method D carries out the same calculations as those of method C in the steps, the coefficients of the polynomials are stored and the calculation units 135₁ through 135_(2t+1) are arranged specially according to method D to reduce the number of registers for storing the polynomial coefficients and the number of calculations in each step to half those of method C.

According to method D, furthermore, the algorithm can be achieved by the (2t+1) calculation units 135₁ through 135_(2t+1) whose number is greater, by 1, than the minimum number 2t of calculations (multiplications) which are required in principle in each step of the Euclidean mutual division method, insofar as each of the calculation units 135₁ through 135_(2t+1) is used only once in the calculations in one step of the Euclidean mutual division method.

The number of registers for storing the coefficients of polynomials required during the calculations is a minimum of 4t+3 in principle. This is because the process of storing the polynomial coefficients according to method D is entirely different from the process of storing the polynomial coefficients according to method C as described above.

More specifically, method C requires registers for independently storing the coefficients of polynomials. Therefore, it is necessary to have (8t+1) registers sufficient to store the coefficients even if each of the polynomials is of the greatest degree. The large number of registers results in an increase in the number of calculation units.

According to method D, as shown in FIG. 14, the A-side registers 141₀ through 141_(2t) are used to store the coefficients of R_(i) (X), μ_(i) (X). In each of the calculation steps, the high-order coefficient (of degree dR_(i)) of the polynomial R_(i) (X) of degree dR_(i) is stored in the register 141₀, and the successive low-order coefficients of R_(i) (X) are stored respectively in successive registers. Following the coefficient of degree 0 of R_(i) (X), the low-order coefficient (of degree 0) of the polynomial μ_(i) (X) is stored in a corresponding register, and the successive high-order coefficients of μ_(i) (X) are stored respectively in successive registers. Therefore, the A-side registers 141₀ through 141_(2t) store, successively from the register 141₀, the coefficients of R_(i) (X) successively from its high-order coefficient, and then the coefficients of μ_(i) (X) successively from its low-order coefficient.

The B-side registers 142₀ through 142_(2t+1) are used to store the coefficients of Q_(i) (X), λ_(i) (X). In each of the calculation steps, the high-order coefficient (of degree dQ_(i)) of the polynomial Q_(i) (X) of degree dQ_(i) is stored in the register 142₀, and the successive low-order coefficients of Q_(i) (X) are stored respectively in successive registers. Following the coefficient of degree 0 of Q_(i) (X), the low-order coefficient (of degree 0) of the polynomial λ_(i) (X) is stored in a corresponding register, and the successive high-order coefficients of λ_(i) (X) are stored respectively in successive registers.

At this time, the degrees of the polynomials satisfy the following relationship:

    dR.sub.i +dμ.sub.i <dQ.sub.i +dλ.sub.i =2t       (43)

Therefore, since the sum of dQ_(i), dλ_(i) is always 2t, the number of the B-side registers 142₀ through 142_(2t+1) may be 2t+2, and the number of the A-side registers 141₀ through 141_(2t) may be 2t+1, which is smaller than the number of the B-side registers by 1.

According to the method shown in FIG. 14, the polynomial R_(i) (X) is shifted to the high-order position (i.e., multiplied by X) each time calculations are carried out in one step of the Euclidean mutual division method.

When the polynomials λ_(i) (X), μ_(i) (X) are inversely stored according to the method D, since the coefficients of these polynomials are stored successively from the low-order coefficient, the polynomial μ_(i) (X) is shifted toward the low-order location as compared with the polynomial λ_(i) (X), achieving the same result as if the polynomial λ_(i) (X) is shifted toward the low-order location as compared with the polynomial μ_(i) (X). At this time, the calculation units are arranged to effect necessary calculations on the polynomials R_(i) (X), λ_(i) (X).

Each of the calculation units 135₁ through 135_(2t+1) shown in FIG. 15 is slightly more complex in arrangement than those required by method C because the data path switchers 136, 138 are added. However, such an addition is very small, and the overall circuit scale can greatly be reduced as the number of calculation units and the number of coefficient-storing registers are reduce to half.

The calculation units 135₁ through 135_(2t+1) may be of any specific arrangement insofar as it can perform the calculations indicated in the tables 2-1, 2-2, 2-3 above.

Consequently, there may be many various arrangements other than the arrangement shown in FIG. 15 for carrying out the same calculations. It is possible to employ a circuit arrangement which is more complex than the circuit arrangement shown in FIG. 15 for operation at higher speed.

According to method D, there are various methods of initializing the registers 141₀ through 141_(2t), 142₀ through 142_(2t+1), 143, 144. If selectors are connected to the input terminals of the registers 141₀ through 141_(2t), 142₀ through 142_(2t+1), 143, 144, then it is possible to initialize all the registers at the same time by shifting the selectors to initial value setting units and inputting values (initial values) set by the initial value setting units to the registers 141₀ through 141_(2t), 142₀ through 142_(2t+1), 143, 144.

Alternatively, the registers can be successively initialized by serially inputting initial values of respective polynomials as inputs to selectors connected to the input terminals of only low-order registers 141_(2t), 142_(2t+1) for respective polynomials.

Generally, the required operating speed of error-correcting circuits varies depending on the application in which it is used. In an application as a microprocessor peripheral which demands a low operation speed, an error-correcting circuit is required to have a minimum circuit scale while maintaining a certain operation speed. When used with a high-speed video signal processing device, an error-correcting circuit is required to process data which is continuously input.

The largest circuit scale of all error-correcting circuit arrangements is needed to implement a process for determining an error locator polynomial from syndromes. Either the method A or B or C or D employs the Euclidean mutual division method for performing such a process. The method D according to the first embodiment, which is better in principle than the methods A, B, C, achieve the algorithm with (2t+1) calculation units whose number is greater, by 1, than the minimum number 2t of calculations which are required in principle in each step of the revised Euclidean mutual division method (2).

According to the method D, the calculations in one step are completed in one clock cycle. Therefore, desired σ(X), ω(X) are obtained after 2t clock cycles.

In processing data that is continuously input, i.e., in highest-speed processing, the process for determining an error locator polynomial from syndromes may be effected during a period of time in which data of one code length is input. Usually, a code length n is sufficiently large compared with 2t. Therefore, if the circuit according to method D is used with respect to an error-correcting system for a sufficiently large code length n, the circuit for carrying out the Euclidean mutual division method will be idling during (n-2t) clock cycles, and hence the circuit is of a redundant arrangement.

In processing data that is not continuously input, the circuit arrangement according to method D is not required to obtain an answer in 2t clock cycles though the processing may be finished within a predetermined period of time.

In view of the above problems, it is necessary to reduce the circuit scale by repeatedly using a small number of calculation units within a required period of time.

It is particularly desirable to meet such a requirement by improving the method D which requires the smallest circuit scale in principle.

According to a second embodiment of the present invention, an Euclidean mutual division circuit has a reduced number of calculation units with an increased degree of multiplexing, resulting in a large reduction in the circuit arrangement, and high-speed operation for a greatly increased throughput.

In the second embodiment, the method D described above according to the first embodiment is improved to use a reduced number of calculation units efficiently repeatedly.

As described above, 2t calculations (multiplications) are effected on the coefficients of polynomials R_(i-1) (X), Q_(i-1) (X) , μ_(i-1) (X), λ_(i-1) (X) in one step of the revised Euclidean mutual division method (2).

According to the method D, the 2t calculations are performed by (2t+1) calculation units so that the calculations in one step are finished in one clock cycle. Thus, the (2t+1) calculation units carry out only one calculation with respect to the calculations in one step of the revised Euclidean mutual division method (2).

It is assumed that t=2 symbols are error-corrected using a finite field GF(2⁴) that is defined using an irreducible polynomial g(X)=X⁴ +X+1. The syndrome polynomial S (X) is defined as:

    S(X)=α.sup.8 X.sup.3 +α.sup.10 X.sup.2 +α.sup.5 X+α.sup.12                                          (44)

This example is the same as the example used above.

                  TABLE 5-3                                                        ______________________________________                                         [STEP 3]                                                                       Cross mode dR = 2, dQ = 3                                                                           Calculations                                                                               Calculations                                  i    A.sub.i B.sub.i on A side   on B side                                     ______________________________________                                         0    α.sup.6                                                                          α.sup.8                                                                          S = B/A = α.sup.2                                                                    --                                            1    α.sup.9                                                                          α.sup.10                                                                         A = B + α.sup.2 × A                                                            B = A                                         2    α.sup.6                                                                          α.sup.5                                                                          A = B + α.sup.2 × A                                                            B = A                                         3    1       α.sup.12                                                                         A = B       B = A                                         4    0       α.sup.9                                                                          A = B       B = A + α.sup.2 × B               5    0       α.sup.7                                                                          A = B       B = A + α.sup.2 × B               ______________________________________                                    

                  TABLE 5-4                                                        ______________________________________                                         [STEP 4]                                                                       Normal mode dR = 2, dQ = 2                                                                          Calculations                                                                               Calculations                                  i    A.sub.i B.sub.i on A side   on B side                                     ______________________________________                                         0    α.sup.14                                                                         α.sup.6                                                                          S = A/B = α.sup.8                                                                    --                                            1    α.sup.4                                                                          α.sup.9                                                                          A = A + α.sup.8 × B                                                            B = B                                         2    α.sup.12                                                                         α.sup.6                                                                          A = A + α.sup.8 × B                                                            B = B                                         3    α.sup.9                                                                          1       A = A       B = B + α.sup.8 × A               4    α.sup.7                                                                          α.sup.11                                                                         A = A       B = B + α.sup.8 × A               5    0       α.sup.9                                                                          A = A       B = B + α.sup.8 × A               ______________________________________                                    

As is clear from the tables 5-3 and 5-4, according to method D, the coefficients stored in the registers whose register number i ranges from 1 to 2t+1, i.e., the coefficients ranging from 1 to 5, are calculated in the five calculation units 135₁ through 135₅.

In the method according to the second embodiment, one calculation unit is used in a time-division multiplexing fashion for carrying out calculations in one step of the revised Euclidean mutual division method (2) thereby to reduce the number of calculation units required. Consequently, although the number of calculation clock cycles is increased, it is possible to reduce the number of calculation units, and to reduce the overall circuit scale.

The degree L_(opt) of multiplexing per calculation unit may be any integer ranging from 1 to 2t. Since if L_(opt) =1, then the number of calculation units required is the same as the number of calculation units according to method D, a circuit arrangement where the degree of multiplexing is 2 or more, i.e., L_(opt) ≧2, will be considered below.

An Euclidean mutual division circuit according to the second embodiment is schematically shown in FIG. 21.

As with method D, the Euclidean mutual division circuit basically comprises a DIV block 1 and an MLT block 2. The DIV block 1 comprises a single division unit 3 for a high-order coefficient, and a determination and control unit 4 for detecting the magnitudes of dR_(i), dQ_(i) and also detecting whether the coefficient of degree 0 of R_(i) (X) to generate a signal for controlling control modes of calculation units 91 through 9_(Kopt) and data path switchers 17₁ through 17_(Kopt).

Values stored in a register 5 for DR and a register 6 for DQ in the DIV block 1 are indicative of the degrees of coefficients of polynomials R_(i) (X), Q_(i) (X) stored in A- and B-side registers 70, 80. When actual calculations are carried out in cross and normal modes (described later on), these stored values indicate the degrees of R_(i) (X) , Q_(i) (X).

The determination and control unit 4 in the DIV block 1 determines an operation mode from the results of comparison of DR, DQ stored in the registers 5, 6 and the result of detection as to whether the value is 0 from the register 70. Based on the determined operation mode and the value of DR from the register 5, the calculation units 9₁ through 9_(Kopt) in the MLT block 2 are controlled independently of each other.

The MLT block 2 is composed of the calculation units 9₁ through 9_(Kopt) each composed of a multiplier, an adder, and data path switchers, a plurality of data path switchers 17₁ through 17_(Kopt), and a plurality of registers 7₁ through 7_(2t), 8₁ through 8_(2t+1) for storing coefficients. The overall system employs Kop_(t) calculation units.

The coefficients of polynomials to be calculated are stored in the same manner as with the method D. That is, storage locations for the coefficients of Q_(i) (X), λ_(i) (X) and storage locations for the coefficients of R_(i) (X), μ_(i) (X) are used respectively in common.

To implement a system capable of error-correcting t symbols according to the second embodiment, there are required registers for storing the coefficients of polynomials R_(i) (X), Q_(i) (X), μ_(i) (X), λ_(i) (X).

Of the registers 7₀ through 7_(2t), 8₀ through 8_(2t+1) shown in FIG. 21, the lefthand group of registers 7₀ through 7_(2t) are referred to as A-side registers, and the righthand group of registers 8₀ through 8_(2t+1) are referred to as B-side registers. In the DIV block 1 and the MLT block 2, there are a total of (2t+1) A-side registers and a total of (2t+2) B-side registers, the total number of the registers being 4t+3. Therefore, the total number of registers used is the same as the total number of registers according to the method D.

In the calculation in each step of the revised Euclidean mutual division method (2), the relationship expressed by:

    dR.sub.i +dμ.sub.i <dQ.sub.i +dλ.sub.i =2t       (45)

is satisfied. This relationship indicates that the sum of the degrees of Q_(i) (X), λ_(i) (X) is always 2t. Therefore, even if the degree of λ_(i) (X) increases as the degree of Q_(i) (X) decreases, the sum of the numbers of registers for storing the coefficients of polynomials is 2t+2 at all times. Similarly, since the sum of the degrees of R_(i) (X), μ_(i) (X) is 2t-1 or smaller, the sum of the numbers of registers for storing the coefficients may be at most 2t+1.

The registers 7₀ through 7_(2t) shown in FIG. 21 are registers for storing the coefficients of R_(i) (X), μ_(i) (X) .

In each calculation step of the revised Euclidean mutual division method (2), the coefficients of the polynomial R_(i) (X) of degree dR_(i) are stored in the registers 7₀, 7₁, 7₂, . . . successively from the high-order coefficient (of degree dR_(i)).

Following the coefficient of degree 0 of R_(i) (X), the coefficients of μ_(i) (X) are stored successively from the low-order coefficient (of degree 0). Therefore, the coefficients of R_(i) (X) are stored successively from the high-order coefficient in the A-side registers 7₀ through 7_(2t) successively from the high-order register, and then the coefficients of μ_(i) (X) are stored successively from the low-order coefficient in the A-side registers.

The registers 8₀ through 8_(2t+1) shown in FIG. 21 are registers for storing the coefficients of Q_(i) (X), λ_(i) (X).

In each calculation step, the coefficients of the polynomial Q_(i) (X) of degree dQ_(i) are stored in the registers 8₀, 8₁, 8₂, . . . successively from the high-order coefficient (of degree dQ_(i)). Following the coefficient of degree 0 of Q_(i) (X), the coefficients of λ_(i) (X) are stored successively from the low-order coefficient (of degree 0). Inasmuch as dQ_(i) +dλ_(i=2t) and the sum of the degrees is constant at all times, the coefficients of Q_(i) (X) are stored in the B-side registers 8₀ through 8_(2t+1) successively from the high-order register, and then the coefficients of λ_(i) (X) are stored in the B-side registers successively from the low-order register.

A total number K_(opt) of calculation units 9₁ through 9_(Kopt) are provided one for L_(opt) sets of registers. Each of the calculation units 9₁ through 9_(Kopt) carries out (2t+1) calculations L_(opt) times which are required in each of the steps shown in the above tables 5-3 and 5-4.

More specifically, the calculation unit 9₁ having a calculation unit number 1 calculates the set of coefficients stored in the registers 7₁ through 7_(Lopt), 8₁ through 8_(Lopt) having register numbers 1 through L_(opt) in L_(opt) clock cycles per step of the Euclidean mutual division method (2). That is, it effects calculations on the set of coefficients stored in the registers 7₁ through 7_(Lopt), 8₁ through 8_(Lopt) having register numbers 1, 2, . . ., L_(opt) in 1, 2, . . ., L_(opt) clock cycles in one step.

Likewise, the calculation unit 9₂ having a calculation unit number 1 affects calculations on the set of coefficients stored in the registers 7_(Lopt+1) through 7₂.Lopt, 8_(Lopt+1) through 8₂.Lopt having register numbers L_(opt+1), L_(opt+2), . . ., 2.L_(opt) in 1, 2, . . ., L_(opt) clock cycles.

The calculation units 9₁ through 9_(Kopt) are connected in cascade in combination with the registers 7₀ through 7_(2t), 8₀ through 8_(2t+1), allowing the overall system to perform (2t+1) calculations per step in L_(opt) clock cycles.

The entire system has (2.L_(opt).k_(opt) +1) registers 7₀ through 7_(2t), 8₀ through 8_(2t+1) for storing coefficients.

Depending on the selected degree L_(opt) of multiplexing, the number of registers may be slightly larger than the number 4t+3 of registers which are essentially necessary. In this case, the value stored in the register 8_(2t+1) which is handled by the final calculation unit 9_(Kopt) is 0 at all times, and no calculation is necessary therefor. By slightly modifying the method of correcting the register for the final calculation unit 9_(Kopt), therefore, it is possible to reduce the number of registers to the minimum register number 4t+3.

[Initialization ]

Setting the registers to initial values will be described below. Initial values for dR₀, dQ₀, dλ₀, dμ₀ which indicate the respective degrees of R₀ (X), Q₀ (X), λ₀ (X), μ₀ (X) are considered to be 2t-1, 2t, 0, -1, respectively. The initial value for the degree dμ₀ of μ₀ (X) is selected to be -1 for the convenience of the algorithm.

Therefore, 2t coefficients (syndromes) of R₀ (X)=S(X) are stored in the registers 7₀ through 7_(2t+1). Since the degree of μ₀ (X) is -1 for the convenience of the algorithm, the number of coefficients of μ₀ (X) is 0, i.e., there is no corresponding register, so that the register 7_(2t) is set to 0.

As regards to the initial values for the registers 8₀ through 8_(2t), since Q₀ (X)=X_(2t), 1 is stored in the register 8₀, and 0 is stored in the registers 8₁ through 8_(2t). As λ₀ (X)=1, 1 is stored in the register 8_(2t+1).

Initial values 2t-1 and 2t which represent the degrees of R_(i) (X) , Q_(i) (X) are stored in the registers 5, 6 which store dR_(i), dQ_(i), respectively. A circuit for setting initial values is required in addition to the circuit arrangement shown in FIG. 21. However, such a circuit is omitted from illustration as it is not essential and is a simple circuit.

[Operation]

The initial values in registers 7_(i) through 7_(LOPT) and 8₁ through 8_(LOPT) are calculated in the first embodiment and then the values of the registers 5 and 6 are compared by an operation mode determination and control circuit 10. A 0-detecting circuit 11 detects whether the register 7₀ which stores the coefficient (A₀) of degree DR of R_(i-1) (X) is 0.

The operation mode determination and control circuit 10 recognizes a cross mode when DR <DQ and A₀ ≠0, and a normal mode when DR>DQ. When DR<DQ and A₀ ≠0, the register 5 does not indicate the degree of R_(i-1) (X), and no calculations can be carried out in the cross mode, so that a shift mode is selected. The determined operation mode is maintained in the L_(opt) clock cycles.

When the operation mode is determined, data path switchers 12, 13 in the DIV block 1 switches to select crossed data or normal data depending on the determined operation mode. The value of the register 7₀, i.e., the high-order coefficient (of degree dR_(i-1)) of R_(i-1) (X), and the value of the register 8₀, i.e., the high-order coefficient (of degree dQ_(i-1)) of Q_(i-1) (X) pass through the data path switcher 13 to a divider 14. The divider 14 effects a division F/G on its inputs F, G, and outputs a result S.

A register 15 holds the value S in the L_(opt) clock cycles which are a calculation time in one step.

Each of the calculation units 9₁ through 9_(Kopt) in the MLT block 2 carries out calculations depending on the operation mode determined by the DIV block 1. The calculations themselves are the same as those according to the method D. According to the method of the second embodiment, the set of coefficients stored in all the registers 7₀ through 7_(2t), 8₀ through 8_(2t+1) is calculated not in one clock cycle, but L_(opt) clock cycles, in one step. Consequently, the same values as those produced upon completion of one clock cycle (i.e., one step) according to the method D are stored in the registers 7₀ through 7_(2t), 8₀ through 8_(2t+1) for the first time upon elapse of the L_(opt) clock cycles.

Each of the calculation units 9₁ through 9_(Kopt) has input terminals A, B and output terminals A', B'. In the ith step, the coefficients stored in the registers 7_(j), 8_(j) having a register number j are calculated as indicated by tables 6-1 through 6-3 given below. These calculations are identical to those according to the method D.

                                      TABLE 6-1                                    __________________________________________________________________________     Calculation table in the cross mode                                            Cross mode                                                                     Calculation  Calculation                                                       unit number  mode  Calculations                                                __________________________________________________________________________     0 < j ≦ dR.sub.i-1                                                                   1     A' = A × S + B                                                                      B' = A                                           dR.sub.i-1 < j ≦ dQ.sub.i-1                                                          2     A' = B     B' = A                                           dQ.sub.i-1 < j                                                                              3     A' = B     B' = B × S + A                             __________________________________________________________________________

                                      TABLE 6-2                                    __________________________________________________________________________     Calculation table in the normal mode                                           Normal mode                                                                    Calculation                                                                               Calculation                                                         unit number                                                                               mode    Calculations                                                __________________________________________________________________________     0 < j ≦ dQ.sub.i-1                                                                 4       A' = B × S + A                                                                    B' = B                                             dQ.sub.i-1 < j ≦ dR.sub.i-1                                                        5       A' = A   B' = B                                             dR.sub.i-1 < j                                                                            6       A' = A   B' = A × S + B                               __________________________________________________________________________

                  TABLE 6-3                                                        ______________________________________                                         Calculation table in the shift mode                                            Shift mode                                                                     Calculation                                                                               Calculation                                                         unit number                                                                               mode          Calculations                                          ______________________________________                                         All j      5             A' = A   B' = B                                       ______________________________________                                    

In one step, the calculations indicated in the above tables 6-1 through 6-3 are carried out in L_(opt) clock cycles using the K_(opt) calculation units 9₁ through 9_(Kopt). The calculation unit 9K having a calculation unit number K effects calculations on the coefficients stored in the registers 7.sub.(K-1).Lopt+1 through 7_(K).Lopt, 8.sub.(K-1).Lopt+1 through 8_(K).Lopt having register numbers (K-1).L_(opt+) ₁ through K.L_(opt).

The coefficients are calculated successively from those stored in registers with younger register numbers in L_(opt) clock cycles. For example, the calculation unit 9₁ calculates the coefficients stored in the registers 7₁, 8₁ with a register number 1 in the first clock cycle in each step, and also calculates the coefficients stored in the registers 7₂, 7₃, . . ., 7_(Lopt), 8₂, 8₃, . . ., 8_(Lopt) with register numbers 2, 3, . . ., L_(opt) in the second, third, . . ., L_(opt) clock cycles.

The coefficients stored in the registers 7₀ through 7_(2t), 8₀ through 8_(2t+1) are shifted successively to registers with younger register numbers in successive clock cycles. Calculations in each clock cycle are performed on the coefficients stored in the high-order registers 7₁, . . ., 7.sub.(K-1).Lopt+1, 8₁, . . ., 8.sub.(K-1).Lopt+ that are handled by the calculation units 9₁ through 9_(Kopt).

The calculated results output from the A side of the calculation units 9₁ through 9_(Kopt) are stored in the A-side registers 7₀, . . ., 7.sub.(K-2).Lopt only in the first clock cycle, and stored in the registers 7_(Lopt-1), . . ., 7.sub.(K-1).Lopt-1 in the second through L_(opt) th clock cycles. The calculated results are shifted successively toward the high-order register in each clock cycle. The registers 7_(Lopt-1), . . ., 7.sub.(K-l).Lopt-1 store data from the registers 7_(Lopt), . . ., 7.sub.(K-1).Lopt only in the first clock cycle, and store the outputs from the A side of the calculation units 9₁ through 9_(Kopt) in the second through L_(opt) th clock cycles. The data path switchers 17₁ through 17_(Kopt) serve to control the storage of the data.

The calculated results output from the B side of the calculation units 9₁ through 9_(Kopt) are stored in the registers 8_(Lopt), 8₂.Lopt, . . ., 8.sub.(K-2).Lopt, and shifted successively.

The registers 7₀, 8₀ store the results of calculations in the first clock cycle, and hold the stored results during the second through L_(opt) th clock cycles.

The registers 7_(Lopt), 7₂.Lopt, 7₃.Lopt, . . . disposed in the junctions between the calculation units 9₁ through 9_(Kopt) store the results of calculations output from the calculation units 9₂ through 9_(Kopt) only in the first clock cycle, and hold the stored result during the second through L_(opt) th clock cycles.

The calculation units 9₂ through 9_(Kopt) operate in the same manner. In each clock cycle, the calculation units 9₂ through 9_(Kopt) determine calculations to be executed, using the tables 6-1 through 6-3, from parameters including a calculation unit number K, a register number L storing data to be calculated, the value in the register DR, and an operation mode.

Using the above calculation process and the process of storing data in the registers 7₀ through 7_(2t), 8₀ through 8_(2t+1), the values obtained after the L_(opt) clock cycles when the calculations in one step are finished are the same as the values stored in the registers when the calculations in one step are finished according to the method D.

By repeating the procedure for finishing the calculations in one step 2t times in the L_(opt) clock cycles, desired σ(X), ω(X) can be obtained.

The calculation units 9₁ through 9_(Kopt) may be of the same arrangement as the calculation units according to the method D. That is, each of the calculation units 9₁ through 9_(Kopt) may be composed of three data path switchers 20 through 22, a multiplier 25, and an adder 26 as shown in FIG. 22.

The detailed arrangement of the calculation units 9₁ through 9_(Kopt), the calculations in the shift mode, and the process for storing data in the registers may be identical to those which have been described above with respect to the method D.

Operation of the Euclidean mutual division circuit shown in FIG. 21, which is composed of the calculation units 9₁ through 9_(Kopt) shown in FIG. 22, is shown by way of example in FIGS. 23 through 35. The illustrated example is the same as the example described above in which t=2 symbols are error-corrected using a finite field GF(2⁴) that is defined using an irreducible polynomial g(X)=X⁴ +X+1.

In this example, the degree L_(opt) of multiplexing is set to 3. Therefore, with the calculations in one step being carried out in 3 clock cycles, the calculations in 2t =4 steps are completed in 2t.L_(opt) =12 clock cycles.

The number K_(opt) of calculation units is given by: ##EQU9##

It is assumed that all the registers 7₀ through 7₅, 8₀ through 8₆ shown in FIG. 23 have been initialized.

In a first step shown in FIGS. 23 through 25, since the value (dR₀) of DR stored in the register 5 is 3, and the value (dQ₀) of DQ stored in the register 6 is 4, the condition DR<DQ is satisfied, and the coefficient of degree dR₀ of R₀ (X) stored in the register 141₀ is α⁸ which is nonzero. Therefore, the operation mode is determined as the cross mode. The data path switcher 13 on the input side of the divider 14 is switched into the cross state, and the data path switcher 12 on the input side of the registers 5, 6 is switched into the cross state. The divider 14 carries out a calculation 1/α⁸. The result S=α⁷ of division is output and stored in the register 15.

When the operation mode in the first step is determined as the cross mode, the operation mode and the result S of division stored in the register 15 are maintained during L_(opt) =3 clock cycles in which the calculations in the first step are carried out.

According to the table 6-1, the data stored in the registers whose register number is j are calculated in one of three different calculation modes depending on whether the register number j is in the range of 0<j ≦dR_(i-1), dR_(i-1) <j≦dQ_(i) -1, or dQ_(i) -1<j.

In the first step shown in FIGS. 23 through 25, since dR₀ =3, dQ₀ =4, the data stored in the registers 7₁ through 7₃, 8₁ through 8₃ whose register numbers are 1 through 3 are subjected to calculations indicated by A'=A×S+B and B'=A. For such calculations, in the calculation unit 9₁, the data path switcher 20 is in the cross state, the data path switcher 21 in the normal state, and the data path switcher 22 in the through state.

The data stored in the registers 7₄, 8₄ whose register number is 4 are subjected to calculations indicated by A'=B and B'=A. For such calculations, in the calculation unit 9₂, the data path switcher 20 is in the cross state, the data path switcher 21 in the normal state, and the data path switcher 22 in the clear state.

The data stored in the remaining registers 7₅, 8₅, 8₆ whose register numbers are 5 and 6 are subjected to calculations indicated by A'=B and B'=B×S+A. For such calculations, in the calculation unit 9₂, the data path switcher 20 is in the normal state, the data path switcher 21 in the cross state, and the data path switcher 22 in the through state.

In the first step, the first clock cycle shown in FIG. 23, the calculation units 9₁, 9₂ carry out calculations the coefficients stored in the registers 7₁, 7₄, 8₁, 8₄ whose register numbers are 1 and 4. Since the calculation unit 9₁ effects calculations indicated by A'=A×S+B, B'=A, the data path switcher 20 is in the cross state, the data path switcher 21 in the normal state, and the data path switcher 22 in the through state. Since the calculation unit 9₂ effects calculations indicated by A'=B, B'=A, the data path switcher 20 is in the cross state, the data path switcher 21 in the normal state, and the data path switcher 22 in the clear state.

The data path switchers 17₁, 17₂ for the calculation units 9₁, 9₂ operate as shown in FIG. 23.

Since FIG. 23 shows the first clock cycle, the data path switchers 17₁, 17₂ are switched into the shift state. The results of calculations output from the output terminals A' of the calculation units 9₁, 9₂ are stored respectively in the registers 7₀, 7₃, and are held thereby for successive 3 clock cycles. Similarly, the register 8₀ stores the value of the register 7₀, and holds the stored value for successive 3 clock cycles.

The results of calculations output from the output terminals B' of the calculation units 9₁, 9₂ are stored respectively in the registers 8₃, 8₆, and are shifted in each clock cycle.

The results of calculations are stored in the registers 7₀ through 7₅, 8₀ through 8₆ as shown in FIG. 24.

The statuses of the registers shown in FIG. 24, which represent the results of calculations illustrated in FIG. 23, are indicative of only the completed calculations on the data stored in the registers 7₁, 8₁ with the register number 1 and the registers 7₄, 8₄ with the register number 4 among the calculations in the first step.

In the calculations following those shown in FIG. 24, i.e., in the first step, the second clock cycle, the calculation units 9₁, 9₂ effect calculations on the coefficients stored in the registers 7₂, 8₂ with the register number 2 and the registers 7₅, 8₅ with the register number

In order for the calculation unit 9₁ to effect calculations indicated by A'=A×S+B, B'=A as in the first clock cycle, the data path switcher 20 is switched into the cross state, the data path switcher 21 into the normal state, and the data path switcher 22 into the through state. In order for the calculation unit 9₂ to effect calculations indicated by A'-B, B'=B×S+A, the data path switcher 20 is switched into the normal state, the data path switcher 21 into the cross state, and the data path switcher 22 into the through state.

Since the clock cycle shown in FIG. 24 is not the first clock cycle, the data path switchers 17₁, 17₂ are switched into a loop state. The results of calculations output from the output terminals A' of the calculation units 9₁, 9₂ are stored respectively in the registers 7₂, 7₅, and are shifted in each clock cycle.

The results of calculations output from the output terminals B' of the calculation units 9₁, 9₂ are stored respectively in the registers 8₃, 8₆ as in the first clock cycle, and are shifted in each clock cycle.

The results of calculations are stored in the registers 7₀ through 7₅, 8₀ through 8₆ as shown in FIG. 25.

The statuses of the registers shown in FIG. 25, which represent the results of calculations illustrated in FIG. 24, indicate the completed calculations on the data stored in the registers 7₁, 7₂, 7₄, 7₅, 8₁, 8₂, 8₄, 8₅ with the register numbers 1, 2, 4, 5 among the calculations in the first step.

The calculations shown in FIG. 25 are those in the first step, the third clock cycle.

The third clock cycle is the final clock cycle in the first step. The calculation unit 9₁ carries out calculations the coefficients stored in the registers 7₃, 8₃ whose register number is 3. Since the calculation unit 9₁ effects calculations indicated by A'=A×S+B, B'=A as in the first and second clock cycles, the data path switcher 20 is in the cross state, the data path switcher 21 in the normal state, and the data path switcher 22 in the through state.

Since the clock cycle shown in FIG. 25 is not the first clock cycle, the data path switchers 17₁, 17₂ are switched into the loop state. The results of calculations are stored respectively in the registers 7₂, 7₅.

The register 8₆ whose register number is 6 stores 0 that is not required to be calculated. Therefore, the data stored in the register 8₆ may not be calculated. The calculation unit 9₁ may simply effect calculations indicated by A'=A, B'=B, and the data path switcher 20 may be in the normal state, the data path switcher 21 in the normal state, and the data path switcher 22 in the clear state. The data 0 which is stored in the register 8₆ which is inserted for the sake of convenience is stored in the register 7₅.

When the statuses shown in FIG. 26 of the registers 7₀ through 7₅, 8₀ through 8₆, which indicate the results of the calculations shown in FIG. 25, are reached, the calculations in the first step are finished. Comparison between FIG. 26 and FIG. 17 which shows the statuses after the first step is finished according to the method D indicates that the values of the registers in FIGS. 26 and 17 are the same as each other.

According to the method D, the values in the A-side registers 141₀ through 141₄ are shifted once toward the high-order location in each clock cycle. According to the method of the second embodiment, however, the values in the A-side registers 7₀ through 7₅ are shifted once for the first time in L_(opt) =3 clock cycles.

The calculations in the second steps are carried as shown in FIGS. 26 through 28. In the second step, since the value (dR₁) in the register 5 is 3 and the value (dQ₁) in the register 6 is 3, with DR≧DQ, the operation mode is recognized as the normal mode. The data path switcher 13 on the input side of the divider 14 is switched into the normal state, and the divider 14 carries out a calculation α² /α⁸. The result S=α⁹ of division is output to the calculation units 9₁, 9₂.

In FIGS. 26 through 28, the calculations are carried out in the normal mode. Consequently, the calculations are carried out according to the calculation table 6-2 in the normal mode.

In the calculations in the second step shown in FIGS. 26 through 28, the data stored in the registers 7₁ through 7₃, 8₁ through 8₃ whose register numbers are 1 through 3 are subjected to calculations indicated by A'=B×S+A and B'=B. For such calculations, in the calculation unit 9₁, the data path switcher 20 is in the normal state, the data path switcher 21 in the normal state, and the data path switcher 22 in the through state.

The data stored in the remaining registers 7₄, 7₅, 8₄, 8₅, 8₆ whose register numbers are 4, 5, and 6 are subjected to calculations indicated by A'=A and B'=A×S+B. For such calculations, the data path switcher 20 is in the cross state, the data path switcher 21 in the cross state, and the data path switcher 22 in the through state.

The calculations on the data in the registers 7₀ through 7₅, 8₀ through 8₆ in the second step are effected separately in three clock cycles. More specifically, the coefficients stored in the registers 7₁ through 7₃, 8₁ through 8₃ whose register numbers are 1, 2, and 3 are calculated successively in each clock cycle by the calculation unit 9₁, and the coefficients stored in the registers 7₄ through 7₆, 8₄ through 8₆ whose register numbers are 4, 5, and 6 are calculated successively in each clock cycle by the calculation unit 9₂.

FIG. 29 shows the values stored in the registers 7₀ through 7₅, 8₀ through 8₆ after the calculations in the second step are finished. It can be seen that the stored values are the same as those values which are stored in the registers after the second step is finished according to the method D as shown in FIG. 18.

The third step shown in FIGS. 29 through 31 is then performed in the cross mode, and the fourth step shown in FIGS. 32 through 34 is performed in the normal mode, until finally polynomials σ(X), ω(X) are obtained as shown in FIG. 35. It should be noted that the coefficients of the polynomials σ(X), ω(X) thus obtained are identical to the coefficients that are obtained according to the method D as shown in FIG. 20.

The method according to the second embodiment carries out exactly the same calculations as those according to the method D. However, by repeatedly using the calculation units 9₁ through 9_(Kopt) in a time-division multiplexing manner with the degree L_(opt) of multiplexing with respect to the calculations in one step, the calculations are carried out with the number K_(opt) =[2t/K_(opt) ]+1 of calculation units which is smaller than the number of calculation units according to the method D. Instead, the number of clock cycles required for the calculations in one step increases to L_(opt).

The method according to the second embodiment is the same as the method D in that the algorithm is achieved by (2t+1) calculations, the number of which is greater, by 1, than the minimum number 2t of calculations (multiplications) required in principle per step of the Euclidean mutual division method. However, the calculations are carried out with the number K_(opt) =[2t/K_(opt) ]+1 of calculation units which is smaller than the number of calculation units according to the method D, by repeatedly using the calculation units in a time-division multiplexing manner with the degree L_(opt) of multiplexing with respect to the calculations in one step.

In the above second embodiment, the degree L_(opt) of multiplexing is L_(opt) =3. If the degree L_(opt) of multiplexing is increased, then the number K_(opt) of calculation units is reduced. When the degree L_(opt) of multiplexing varies, the number L_(opt) of calculation units varies as indicated by the following table 7:

                  TABLE 7                                                          ______________________________________                                         The degree of multiplexing and the number of calculation units                                   Number K.sub.opt of                                          Degree of multiplexing                                                                           calculation units                                            ______________________________________                                         2                 3                                                            3                 2                                                            4                 2                                                            5                 1                                                            ______________________________________                                    

As can be seen from the table 7, as the degree L_(opt) of multiplexing increases, the number L_(opt) of calculation units decreases. Since the number of registers is equal to 4t+3 at all times, the greater the degree L_(opt) of multiplexing, the smaller the circuit scale of the Euclidean mutual division circuit. According to the method of the second embodiment, the circuit scale can greatly be reduced by selecting an optimum degree L_(opt) of multiplexing according to specifications require for an error-correcting system.

The number K_(opt) of calculation units remains the same if the degree of multiplexing is either 3 or 4. Inasmuch as the number of calculation clock cycles is smaller to greater advantage if the degree L_(opt) of multiplexing is smaller, calculations can be carried out with the same number K_(opt) of calculation units in a smaller number of clock cycles when the degree L_(opt) of multiplexing is set to 3.

According to the second embodiment, the number of registers 7₀ through 7₅, 8₀ through 8₆ used is greater, by 2, than the number (4t+3)=11 which is essentially required. This is only for the sake of convenience for an easier understanding of the calculations. Actually, the register 86 whose register number is 6 stores 0 and hence the data stored in the register 8₆ is not required to be calculated. Therefore, by slightly modifying the process of controlling the calculation unit 9₂, it is possible to simply construct the Euclidean mutual division circuit of (4t+3) registers whose number is a minimum number based on the operating principles.

In the method according to the second embodiment, there are various ways of initializing the registers 7₀ through 7_(2t), 8₀ through 8_(2t+1). In this regard, the same description as with the method D applies. More specifically, if selectors are connected to the input terminals of the registers 7₀ through 7_(2t), 8₀ through 8_(2t+1), then it is possible to initialize all the registers at the same time. Alternatively, the registers can be successively initialized by serially inputting initial values of respective polynomials as inputs to low-order registers, i.e., only the registers 7_(2t), 8_(2t+1), for respective polynomials.

Furthermore, the register 15 for storing the result of division from the divider 14 stores the value once in L_(opt) clock cycles. Inputs F, G to be actually divided by the divider 14 are of constant values in L_(opt) - 1 clock cycles before the result of division is stored in the register 15. For example, in FIG. 26, the result α⁹ of division in the normal mode is stored in the register 15, and α² and α⁸ which correspond to the inputs F, G are stored in the registers 7₀, 8₀ at the time of FIG. 24 which is L_(opt) -1 =2 clock cycles before. Therefore, the operation mode in the second step may be determined and the division may be started in the second clock cycle in the first step.

Generally, for the degree L_(opt) of multiplexing, the L_(opt) - 1 clock cycles may be consumed to determine the operation mode and effect the division.

Usually, the speed of divisions is much lower than the speeds of multiplications and additions. According to the method D, the operation speed is governed by the speed of divisions, and the circuit cannot operate at high speed.

According to the method of the second embodiment, however, by the degree L_(opt) of multiplexing to a suitable value matching the speed of operation of the divider 14, then the calculation units 9₁ through 9_(Kopt) can be used efficiently even if the divider 14 is slow. Therefore, even if a slow divider 14 is used, the entire system can operate at high speed.

As described above, the degree of multiplexing may be increased to reduce the number of calculation units for achieving a large reduction in the circuit scale and for high-speed operation for an increased throughput.

Having described preferred embodiments of the invention with reference to the accompanying drawings, it is to be understood that the invention is not limited to that precise embodiments and that various changes and modifications could, be effected by one skilled in the art without departing from the spirit or scope of the invention as defined in the appended claims. 

What is claimed is:
 1. A Euclidean mutual division circuit comprising:a control unit for generating a switching command indicative of one of a normal connection and a cross connection based on a comparison of a predetermined initial value and a value stored in a register in a division unit, and supplying the generated switching command to a plurality of calculation devices and said division unit; said plurality of calculation devices each composed of two registers for storing coefficients of polynomials R_(i) (X), Q_(i) (X), λ_(i) (X), μ_(i) (X) and a calculation unit, said plurality of calculation devices being cascaded in a number corresponding to one more than a minimum number of calculations per step of a Euclidean mutual division, for receiving a result of division and a value output from a preceding stage, and responsive to the switching command for effecting one of a normal-connection calculation and a cross-connection calculation; and said division unit having two registers for storing coefficients of polynomials R_(i) (X), Q_(i) (X), λ_(i) (X) and μ_(i) (X), and a divider, for receiving a value output from a final stage of said plurality of calculation devices, effecting one of a normal-connection division and a cross-connection division in response to the switching command, and supplying the result of division to each of said plurality of calculation devices.
 2. A Euclidean mutual division circuit comprising:a control unit for generating a switching command indicative of one of a normal connection, a cross connection, and a shift connection based on a comparison of a predetermined value and a value stored in a first register in a division unit, and supplying the generated switching command to an MLT block and a division unit; said MLT block having a group of A-side registers divided depending on a degree of multiplexing for storing coefficients of polynomials R_(i) (X) and Q_(i) (X), a group of B-side registers divided depending on said degree of multiplexing for storing coefficients of polynomials λ_(i) (X), μ_(i) (X), and a number of calculation units depending on said degree of multiplexing, said MLT block being capable of receiving a result of division and values output from A-side and B-side registers, and responsive to the switching command for effecting one of a normal-connection calculation, a cross-connection calculation, and a shift-connection calculation during a clock cycle in each step of a Euclidean mutual division; and said division unit having a second register for storing R_(i) (X), Q_(i) (X), a third register for storing λ_(i) (X), μ_(i) (X) and a divider for dividing coefficients stored in the second and third registers, for receiving a value output from a final stage of said MLT block, effecting one of a normal-connection division and a cross-connection division in response to the switching command, and supplying the result of division to each of said calculation units when a step of said Euclidean mutual division is finished.
 3. A Euclidean mutual division circuit according to claim 1 or 2, wherein said division unit includes at least a pair of A-side and B-side registers, and values produced by dividing coefficients stored in said pair of A-side and B-side registers are returned to each of said calculation units.
 4. A Euclidean mutual division circuit according to claim 3, wherein said division unit and said control unit each have at least one switcher for switching connections in said division unit and said control unit in response to said switching command.
 5. A Euclidean mutual division circuit according to claim 4, wherein said result of division represents values of at least one of an error locator polynomial and an error evaluator polynomial.
 6. A Euclidean mutual division circuit according to claim 5, wherein said Euclidean mutual division circuit is used for error-correcting a digital signal, said A-side registers including at least (2t+1) registers and said B-side registers including at least (2t+2) registers, totaling at least (4t+3) registers, where t is the number of symbols that can be error-corrected. 